
Book. ^ 6 / 
Copyright}]^ 

COPYRIGHT DEPOSIT. 



TWENTIETH CENTURY TEXT-BOOKS 




LORD KELVIN (SIR WILLIAM THOMSON) (1824-1907) 

William Thomson ranks as one of the two or three greatest 
physicists of the nineteenth century. He was born in Ireland, but 
spent nearly his entire life in Glasgow, Scotland, where his father 
was professor of mathematics. At the early age of seventeen he 
began to write on mathematical subjects, but his attention was soon 
turned to the study of physics. In 1846 he became professor of 
natural philosophy at Glasgow, where he remained fifty-three years. 

Thomson's early investigations led to the invention of the abso- 
lute scale of temperature. Important experiments in heat were 
carried out by him in collaboration with Joule from 1852-1862. 

Later Thomson became almost universally known by his work 
as electrician of the Atlantic cables, the first of which was laid in 
1858. By the invention of the well-known mirror galvanometer 
which he used in receiving messages, he increased the rate of trans- 
mission from two or three to about twenty-five words per minute. 
This instrument was replaced later by his " siphon recorder," which 
is still in use. 

Thomson was instrumental in establishing the practical system of 
electrical units. He invented the absolute and quadrant electrome- 
ters for measuring potentials, a sounding device for use on moving 
ships, and many other practical measuring instruments. His favorite 
subject, however, was the nature of the ether, which, as he said, 
claimed his attention daily for over forty years. His writings are 
included in his Papers on Electrostatics and Magnetism, Mathemat- 
ical and Physical Papers, Popular Lectures and Addresses, and 
Thomson and Tait's An Elementary Treatise on Natural Philosophy. 



TWENTIETH CENTURY TEXT-BOOKS 



A HIGH SCHOOL COURSE 
IN PHYSICS 



BY 



FREDERICK R. GORTON, B.S., M.A., Ph.D. 

w 
ASSOCIATE PROFESSOR OF PHYSICS, MICHIGAN 

STATE NORMAL COLLEGE 




D. APPLETON AND COMPANY 
NEW YORK CHICAGO 

1910 






COPYKIGHT, 1910, BY 

D. APPLETON AND COMPANY. 



(g, C!,A265301 



PREFACE 

Physics is the summary of a part of human experience. Its devel- 
opment has resulted from the fact that its pursuit has successfully 
met human needs. Hence it is believed that the presentation of the 
subject in the secondary school should be the expansion of the every- 
day life of the pupil into the broader experience and observation of 
those whose lives have been devoted to the study. Human activity 
and progress, therefore, should be the teacher's guiding principle, 
and the bearing of each phenomenon and law on the interests of 
mankind should be clearly disclosed and emphasized. The author 
of this book has accordingly endeavored to give great prominence 
to facts of common observation in the derivation of physical laws, 
and, further, has attempted to point out plainly the service that has 
been afforded mankind by a knowledge of nature's laws. No text- 
book, however, can take the place of the skilled teacher in showing 
clearly the relation of physical phenomena to human activities, or in 
the selection of illustrative examples within the range of observation 
of his pupils. 

Large portions of the subject-matter of Physics deal with knowl- 
edge already possessed by a pupil of high-school age, and nothing 
is of more appealing interest to him than the feeling that this infor- 
mation is to be made of some value. By recalling phenomena well 
within the acquaintance of the pupil and supplementing them with 
demonstrative experiments, the way is easily paved to the deduction 
and interpretation of general principles. In the presentation of such 
experiments, the author has described as simple and inexpensive 
apparatus as he has found to be consistent with satisfactory results. 

No effort has been spared to give the teacher and pupil every pos- 
sible assistance. The sections have been plainly set off and given 
suggestive headings ; references to related material have been inserted 
where needed; and the numerous sets of exercises have been care- 

v 



vi PREFACE 

fully graded. In order to bring the problems near the discussions 
upon which their solutions depend, they have been arranged in more 
and smaller groups than is usual. The exercises throughout the 
book have been selected from concrete cases, and the usual problems 
in pure reductions have been omitted. Illustrative solutions of prob- 
lems and suggestions have been given w^herever difficulties have been 
found to arise. As an aid to the pupil in reviewing and to the 
teacher in conducting rapid drill exercises, a summary of the contents 
of each chapter is presented at its conclusion. 

The educational value of the portraits and biographical sketches 
of many of the great men of science is at once apparent. Emphasis 
upon the parts that these men have played in the development of 
Physics has been recognized by eminent educators as an important 
factor in creating an atmosphere of human interest around the sub- 
ject. These names should become familiar to every student of Physics. 

On account of the rapid advance at the present time in the practi- 
cal uses made of physical principles, the author believes that the 
sections involving new applications will be found of general interest 
and utility. 

The book will be found to be free from the more difficult uses 
of algebraic and geometric principles. The place of first importance 
has been given to the study of phenomena, and mathematical expres- 
sions have been introduced as convenient m^ns of designating exact 
relations which have been previously interpreted. It is mainly in the 
subject of Physics that the pupil is brought to realize the value 
of his mathematical studies in the world of concrete quantities. 

The author believes that the class-room work in the subject should 
be accompanied by a sufficient number of individual laboratory exer- 
cises to fix clearly in mind great principles and important phenomena. 
Further, enough practice with simple drawing instruments should 
be given to enforce the use of the simplest geometrical relations. 

In addition to the many subjects whose treatment is demanded by 
the achievements of recent times, the author has given careful atten- 
tion to the various topics recommended by the Committee of Second- 
ary School Teachers to the College Entrance Examination Board. 

The author desires to acknowledge here the many helpful sugges- 
tions of those who have read and criticised the proof or manuscript, and 



PREFACE vii 

wishes especially to mention Professor E. A. Strong of the Michigan 
State Normal College ; Mr. Fred R. Nichols of the Richard T. Crane 
Manual Training High School, Chicago, 111. ; Mr. Frank B. Spaulding 
of the Boys' High School, Brooklyn; Mr. Albert B. Kimball, Princi- 
pal of the High School, Fairhaven, Mass. ; Professor Karl E. Guthe 
of the University of Michigan ; Mr. George A. Chamberlain, Prin- 
cipal of East Division High School, Milwaukee ; and Mr. J. M. 
Jameson of the Pratt Institute, Brooklyn, N.Y. 



CONTENTS 

CHAPTER I 



Introduction 



CHAPTER II 



Motion, Velocity, and Acceleration 

1. Uniform Motions and Velocities 13 

2. Uniformly Accelerated Motion ..... 16 

3. Simultaneous Motions and Velocities . . 21 
Summary ...... .26 

CHAPTER III 

Laws of Motion — Force 

1. Discussion of Newton's Laws ...... 28 

2. Concurrent Forces ........ 35 

3. Moments of Force — Parallel Forces .... 39 

4. Resolution of Forces ........ 42 

5. Curvilinear Motion ........ 44 

Summary .......... 48 

CHAPTER IV 

Work and Energy 

1. Definition and Units of Work ...... 51 

2. Activity, or Rate of Work 53 

3. Potential and Kinetic Energy 54 

4. Transitions of Energy . . . . . . .58 

Summary .:........ 61 

CHAPTER V 

Gravitation 

1. Laws of Gravitation and Weight . . . . .62 

/ 2. Equilibrium and Stability 65 

ix 



CONTENTS 



3. The Fall of Unsupported Bodies 

4. The Pendulum .... 
Summary ..... 



69 
74 
81 



CHAPTER yi 
Machines 

1. General Law and Purpose of Machines 

2. The Principle of the Pulley 

3. The Principle of the Lever 

4. The Principle of the Wheel and Axle 

5. The Inclined Plane, Screw, and Wedge 

6. Efficiency of a Machine 

Summary ...... 

CHAPTER yil 

Mechanics of Liquids 

1. Forces Due to the Weight of a Liquid 

2. Force Transmitted by a Liquid 

3. Archimedes' Principle 

4. Density of Solids and Liquids . 

5. Molecular Forces in Liquids 
Summary . . . . 



83 
85 
88 
93 
96 
99 
101 



103 
111 
119 
124 
130 
135 



CHAPTER YIII 

Mechanics of Gases 

1. Properties of Gases 

2. Pressure of the Air against Surfaces 

3. Expansibility and Compressibility of Gases 

4. Atmospheric Density and Buoyancy 

5. Applications of Air Pressure . 
Summary ....... 



138 
139 
147 
153 
155 
163 



CHAPTER IX 

Sound : Its Nature and Propagation 

1. Origin and Transmission of Sound 165 

2. Nature of Sound 169 

3. Intensity of Sound ........ 173 

4. Reflection of Sound 176 

Summary ,,.,,,..♦. 177 



CONTENTS xi 
CHAPTER X 

PAGE 

Sound : Wave Frequency and Wave Form 

1. Pitch of Tones 179 

2. Resonance ......... 185 

3. Wave Interference and Beats 188 

4. The Vibration of Strings 192 

5. Quality of Sounds ........ 196 

6. Vibrating Air Columns ....... 198 

Summary 205 

CHAPTER XI 

Heat : Temperature Changes and Heat Measurement 

1. Temperature and its Measurement 208 

2. Expansion of Bodies 216 

3. Calorimetry, or the Measurement of Heat . . . 225 
Summary .......... 229 

CHAPTER XII 

Heat : Transference and Transformation of Heat Energy 

1. Change of the Molecular State of Matter . . . 231 

2. The Transference of Heat 247 

3. Relation between Heat and Work 256 

Summary .......... 265 

CHAPTER XIII 

Light : Its Characteristics and Measurement 

1. Nature and Propagation of Light ..... 268 

2. Rectilinear Propagation of Light ..... 270 

3. Intensity and Candle Power of Lights .... 274 
Summary .......... 277 

CHAPTER XIV 

Light : Reflection and Refraction 

1. Reflection of Light • . . .278 

2. Reflection by Curved Mirrors 284 

3. Refraction of Light 292 



xii CONTENTS 

PAGE 

4. Lenses and Images 300 

5. Optical Instruments 311 

Summary .......... 318 

CHAPTER XV 

Light : Color and Spectra 

1. Dispersion of Light : Color 321 

2. Spectra 324 

3. Interference of Light 330 

Summary .......... 333 

CHAPTER XVI 
Electrostatics 

1. Electrification and Electrical Charges .... 335 

2. Electric Fields and Electrostatic Induction 

3. Potential Difference and Capacity . 

4. Electrical Generators .... 
Summary ....... 



340 
346 
352 
356 



CHAPTER XVII 
Magnetism 

1. Magnets and their Mutual Action ..... 359 

2. Magnetism a Molecular Phenomenon .... 365 

3. Terrestrial Magnetism 368 

Summary .......... 372 

CHAPTER XVIII 

Voltaic Electricity 

1. Production of a Current — Voltaic Cells .... 374 

2. Effects of Electric Currents 388 

Summary .......... 400 

CHAPTER XIX 

Electrical Measurements 

1. Electrical Quantities and Units ..... 402 

2. Electrical Energy and Power ...... 413 

3. Computation and Measurement of Resistances . . 417 
Summary 422 



CONTENTS 



XlU 



CHAPTER XX 

Electro-magnetic Induction 

1. Induced Currents of Electricity 

2. Dynamo-Electric Machinery .... 

3. Transformation of Power and Its Applications 

4. The Telegraph and the Telephone . 
Summary . . . . . . . , 



425 
432 
442 

453 
460 



CHAPTER XXI 
Radiations 

1. Electro-magnetic waves 

2. Conduction of Gases 

3. Radio-activity ..... 



462 
466 
468 



LIST OF PORTRAITS 

Lord Kelvin (Sir William Thomson) . . , Frontispiece 

FACING PAGE 

Sir Isaac Newton 30 

Galileo Galilei 70 

Hermann von Helmholtz ........ 196 

Comit Rumford (Sir Benjamin Thompson) .... 256 ^ 

James Prescott Joule 258 -^ 

Benjamin Franklin . . 346 x 

Count Alessandro Volta_ 352 

Hans Christian Oersted 382- 

Dominique Francois Jean Arago 382 

Andre Marie Ampeje . 406 

George Simon Ohw— 406 

Michael Faraday = . 426 

Joseph Henry . 426 

James Clerk-Maxwell 432 

Heinrich Hertz 464 

Sir William Crookes 466 

Wilhelm Konrad Bontgen 466 

Antoine Henri Becquerel ........ 470 

Madame Curie 470 



CHAPTER I 
INTRODUCTION 

1. Physics. — In the experiences of everyday life we 
witness a great variety of changes in the things around us. 
Objects are moved, melted, evaporated, solidified, bent, 
made hot or cold, and undergo a change in their condi- 
tion, place, or shape in a great many other ways. Physics 
is the science that treats of the properties of different sub- 
stances and the changes that may take place within or 
between bodies, and it investigates the conditions under 
which such changes occur. 

In its broadest sense Physics is the science of phenom- 
ena. Every action of which we become aware through 
the senses is ?i phenomenon. We hear the rolling thunder, 
we see the shining of a live coal, we taste the dissolving 
sugar, we smell tlie evaporating oil, and we feel moving 
air. By considering his own experience the student will 
be able to recall numerous examples of physical phe- 
nomena and to state the sense by means of which he per- 
ceives each of them. 

The study of physics, however, not only directs our 
attention to the phenomena to which we are accustomed, 
but to a multitude of more unusual but not less important 
ones. It also strives to put these phenomena to experi- 
mental tests that will enable us to understand the laws 
connecting actions with their causes. 

2. Utility of the Study of Physics. — Increasing acquaint- 
ance with nature and natural law has been the means of 

2 1 



2 A HIGH SCHOOL COURSE IN PHYSICS 

elevating man from the life of limited power and useful- 
ness of the savage to his condition of present-day enlight- 
enment. The early discovery of fire was a great step 
toward civilization. By some crude experimental study it 
was found later that fire could be produced at will, as by 
the striking of flint and by rubbing two pieces of dry wood 
together. Thus the observation of simple natural phe- 
nomena enabled man to secure heat for cooking his food 
and warming liis habitation, besides aiding him in forming 
implements to procure food, improve his shelter, and give 
him protection from enemies. 

This same observation of natural phenomena has pro- 
duced every existing artificial device for our protection, 
convenience, and comfort. The engineer wlio plans a rail- 
road, with its bridges, tunnels, and grades, together with 
the locomotive and its train, makes use at every step of 
knowledge acquired through the study of Physics. The 
surveyor ascertains how to cut through the hills and fill 
the valleys by the use of instruments which involve physi- 
cal principles. By the discovery and application of physi- 
cal laws scientists and inventors have produced the tel- 
escope, telephone, steam engine, electric car, and all the 
other useful appliances which form so important a part of 
our everyday life. 

3. Matter. — There are three general characteristics by 
which matter is recognized. 

(1) Matter always occupies space. On this account it 
is said to possess the property of extension. Many invisi- 
ble bodies of matter exist. The air in a bottle or tumbler 
is such a body. We ma}^ show, however, by the following 
experiment that it is as real as any other body: 

1. Place a piece of cork upon the surface of water in a vessel, cover 
it with an inverted tumbler, and force the tumbler deep into the 
water as in Fig. 1. The air that was in the tumbler still occupies 



INTRODUCTION 3 

nearly all of that space, and the water is not allowed to rise and 
fiU it. 

(2) All matter is indestructible^ in the sense that it has 
never been discovered that the small- 
est portion can be annihilated by any x^t2/^ H 
process known to man. Causing* a f^^Wj^^^ 
body, as a piece of coal, to disappear ^jf^^nlF'^wl:. \ M 

by burning does not destroy the ma- IfT^jF^^W' ? " 
terial of which it is composed. A | _^^^^^^J|Lj'^^' 

portion is carried away in the smoke ^««««- k«^^ 

and gases, and the remainder left be- Fig. i. — in verted Tumbler 

,.,.,, 1 T -, T,i Nearly Full of Air. 

hind in the ash produced. In the 

process of evaporation a drop of water becomes invisible ; 
but the matter still exists in the atmosphere as a trans- 
parent vapor. 

(3) All matter has weight, i.e. is attracted by the 
earth. In a more general sense it may even be said that 
every body has an attraction for, or pulls upon, every 
other body. The term gravitation is used to express this 
characteristic of matter. 

The fact that air possesses weight as well as extension 
may be shown by experiment as follows : 

2. Remove the brass fixture from an incandescent lamp bulb, and 
carefully balance the bulb on a sensitive beam balance. Introduce a 
short nail into the stem of the bulb, and tap lightly with a hammer 
until the glass is broken and air admitted. If the bulb is now placed 
upon the balance, a decided increase in weight will be observed. 
The weight of air admitted has been added to that of the bulb, which 
originally contained almost no air. 

77ie quantity/ of matter in a body is called its mass. Dif- 
ferent kinds of matter, as gold, water, glass, air, mercury, 
salt, hydrogen, etc., are called substances. Substances are 
recognized by their properties ; as, hardness, elasticity, 
tenacity, fluidity, transparency, etc. 



4 A HIGH SCHOOL COURSE IN PHYSICS 

4. Measurement of Quantities. — A little reflection will 
show the necessity of having systems of measurement for 
the various quantities that we find in nature, such as 
length, area, volume, weight, time, etc. The importance 
of such systems has been recognized by the governments 
of civilized countries, and the values of the units employed 
have been fixed by law. Thus the foot, pound, second, 
etc., are well-established units of quantity, and are in gen- 
eral use throughout Great Britain and the United States. 

The English system of measurement, however, is objec- 
tionable on account of the inconvenient relations between 
the units and their multiples and divisions. For example, 
1 pound = 16 ounces = 7000 grains ; or 1 mile = 320 rods 
= 5280 feet = 63,360 inches. It is mainly for this reason 
that many countries have adopted the metric system of 
measurement, in which the relations are always to be 
expressed by some power of ten. This system greatly re- 
duces the effort required in making correct computations. ^ ^ 
Since in the United States both the English and the 
metric system are employed, it will be advisable to be- 
come proficient in the use of each. 

5. Measures of Extension. — Every body occupies space 
of three dimensions : length, breadth, and thickness ; but 
each of these is simply a lengthy the metric unit of which 
is the meter. The meter is the distance between two 
transverse lines ruled on a platinum-iridium bar kept in 
the Archives at Sevres, near Paris. ^ Qn account of the 

1 Congress recognized the desirability of introducing the metric system 
as early as 1866. See Congressional Globe, Appendix, Part 5, p. 422, 
Chap. CCCI : An act to authorize the use of the metric srjstem of weights 
and measures. By this act the yard is defined as f |§^ of a meter. 

2 The meter was originally intended to be one ten-millionth of the dis- 
tance from the equator to the north pole. Accurate copies of the meter 
and other metric units are kept in the U. S. Bureau of Standards at 
Washington, D.C. 



INTRODUCTION 5 

changes in the length of the bar v/ith variations of tem- 
perature, the distance must be taken when the bar is at 
the temperature of freezing water. The multiples and 
divisions of the meter are designated by prefixes signify- 
ing the relations which they bear to the unit. The multi- 
ples have the Greek prefixes, deka (ten), heeto (hundred), 
kilo (thousand), and myria (ten thousand). The divisions 
have the Latin prefixes, deci (tenth), centi (hundredth), 
and milli (thousandth). The relations are shown in the 
following table : 

Metric Table of Length 

^ A myriameter equals 10,000 meters. 

A kilometer (km.) equals 1,000 meters. 

1 A hectometer equals 100 meters. 

1 A dekameter equals 10 meters. 

A decimeter (dm.) equals 0.1 of a meter. 

A centimeter (cm.) equals 0.01 of a meter. 

A millimeter (mm.) equals 0.001 of a meter. 

The most important equivalents in the English system 
are the following : 

1 meter (m.) equals 39.37 inches, or 1.094 yards. 

2 1 centimeter equals 0.3937 of an inch. 

1 kilometer equals 0.6214 of a mile. 

The metric equivalents most frequently used are : 

1 yard equals 91.44 centimeters, or 0.9144 m. 
2 1 inch equals 2.540 centimeters. 
1 mile equals 1.609 kilometers. 

The relative sizes of the inch and the centimeter are 
shown in Fig. 2. 

Because the meter is too large for general use in Physics, 

the centimeter has been chosen as the unit. The centimeter 

and the gram^ which is the metric unit of mass, and the 

second as a time unit, are together the fundamental units 

1 Seldom used. 2 To be memorized. 



6 



A HIQH SCHOOL COURSE IN PHYSICS 



S 
o 

o 






as 



00 



1 Square 
Centimeter 



of the so-called centimeter-gram-second (C. G. S.) system 
of measurement which is in use throughout the world in 
scientific work. 

6. Surface Measure. — The unit of sur- 
face, or area, in the C. 
G. S. system is the square 
centimeter (cm. 2). It is 
the area of a square whose 
edge is one centimeter in 
length. One square 
inch is thus obviously 
equal to (2.540)2, or 
6.4516 square centi- 
meters. The relative 
E^- sizes of these two units are shown in Fig. 3. 



CO 



lo 



'* 



CO 



(M 



CO 



1 Squarelmh 



Fig. 3. — Relative 
Sizes of the Square 
Inch and Square 
Centimeter. 



c<j 



7. Cubic Measure. - 
in the C. G. S. system 
is the cubic centimeter 
(cm.^). This unit is 
defined as the vol- 
ume of a cube whose 
edge is one centi- 
meter in length. 
One cubic inch thus 



The unit of volume 



equals 
16.387 
meters. 



(2.540)3, or 



iCubie ' 
Centi/pfitpr 



1 Cu die Inch 



Fig. 4, — Relative Sizes 
of the Cubic Inch and 
Cubic Centimeter. 



cubic centi- 
The relative 
sizes of these units are shown in Fig. 4. 
Fig. 2. — Showing 8. Measurcs of Capacity. — The unit of 

the Relative Sizes •, • .i , • , • j^i t^ 

of the English Capacity m the metric system is the liter 
Inch and the Met- Q)i>onounced lecHcr'), which is equal in 

ric Centimeter. . ^ ■ -i - i i 

Size to a cubic decimeter, or one thousand 
cubic centimeters. The liter is somewhat larger than the 
liquid quart and smaller than the dry quart. More pre- 



INTRODUCTION 7 

cisely, a liter equals 1.057 liquid quarts and 0.908 of a 
dry quart. Multiples and divisions of the liter are des- 
ignated by the prefixes explained in § 5, but are little 
used in ordinary physical measurements. 

EXERCISES 

1. The distance from Detroit to Chicago is 280 mi. What is the 
metric equivalent of this distance? 

2. Which is the lower price for silk, $1 per yard or |1.10 per meter ? 
How much is the difference ? 

3. A tourist while in Paris pays the equivalent of 50 ct. per meter 
for cloth worth 40 ct. per yard. How much is the loss on a purchase 
of 20 m.? 

4. If the cost of water is 10 ct. per thousand gallons, what is the 
equivalent cost per cubic meter? (1 gal. = 231 cu. in.) 

5. How much dearer in Germany is oil costing the equivalent of 
5 ct. per liter than the same in the United States at 15 ct. per gallon? 
Express the result in cents per gallon. 

6. How much more cloth will $1 buy at 20 ct. per meter than 
at 18 ct. per yard when the width is 30 in. ? Express the result in 
square inches. 

7. If a railroad ticket in France costs the equivalent of $19.50 per 
thousand kilometers, what is the rate per mile? 

8. If illuminating gas in Germany is sold at the equivalent of 
3.5 ct. per cubic meter, what is the corresponding price per thousand 
cubic feet ? 

9. Measures of Mass. — The unit of mass in the metric 
system is the kilogram^ and in the C. G. S. system, the 
gram. The gram-mass is the one-thousandth part of the 
mass of a standard platinum-iridium cylinder preserved in 
the Archives of France. The entire mass of this cylinder 
is one kilogram (abbreviated to kilo^ pronounced kee'lo^. 
This standard kilogram was intended to be equal to the 
mass of one thousand cubic centimeters, or one liter, of 
pure water; in fact, it may be considered so without ap- 
preciable error. 

This relation between mass and volume in the metric 



8 



A HIGH SCHOOL COURSE IN PHYSICS 



system is of great convenience in physics. Since one 
cubic centimeter of water has a mass of one gram, if we 




Fig. 5. — Relation of the Unit of Mass to the Unit of Volume. 

know the mass in grams of a certain volume of water, the 
volume also is known, and vice versa. (See Fig. 5.) 

Metric Table of Mass 

^ A myriagram equals 10,000 grams (g.)- 

(kg.) equals 1,000 grams. 

equals 100 grams. 

equals 10 grams, 
(dg.) equals 0.1 of a gram, 

(eg.) equals 0.01 



of a gram. 



A kilogram 
1 A hectogram 
1 A dekagram 

A decigram 

A centigram. 

A milligram (mg.) equals 0.001 of a gram. 

The English unit of mass used in Physics is the avoir- 
dupois pound containing 7000 grains and defined as being 
g-gVie ^^ ^ kilogram. Its multiple is the ton, or 2000 
pounds ; its divisions, the ounce and the grain. 

The English and metric equivalents most frequently 
used are as follows : 

1 pound is equal to 453.59 grams. 

1 ounce is equal to 28.35 grams. 

2 1 kilogram is equal to 2.20 pounds. 

1 gram is equal to 15.43 grains. 



1 Seldom used. 



2 To be memorized. 



INTRODUCTION 9 



EXERCISES 



1. Express the mass of a cubic inch of water in grams. 

2. What is the mass of a cubic decimeter of water in pounds? 

3. Sugar at 6 ct. per pound costs how much per kilogram ? 

4. A cubic centimeter of mercury has a mass of 13.6 g. Find the 
mass of a cubic inch of mercury in ounces. Ans. 7.86 oz. 

5. How many pounds are there in a cubic foot of water? 

6. How many grams of water will be required to fill a rectangular 
vessel measuring 20 x 25 x 30 cm. ? Reduce to pounds. 

7. One cubic centimeter of iron has a mass of 7.5 g. Find the 
mass of an iron plate 150 cm. square and 2 mm. thick. 

8. An empty flask weighs 100 g. ; when filled with water, the entire 
mass is 365 g. What is the capacity of the flask ? 

9. If the mass of a given volume of gold is 19 times that of an 
equal volume of water, what is the mass of 25 cm.^ of gold? What is 
the volume of a gold body whose mass is 10 g. ? 

10. Mass Distinguished from Weight. — Mass and weight 
must not be regarded as synonymous terms. If it were 
not for the fact that any two masses attract each other 
(§ 3), we should have little use for the word iveight. 
This attraction between common bodies is beyond our 
power to detect by ordinary means, because it is so slight. 
When, however, one of the attracting bodies is massive, 
as the earth, and the other is some object, as a stone, the 
attraction is great enough to be easily perceived as we 
try to support or lift the smaller body. This downward 
pull of the earth upon the stone is called the weight of the 
stone. The weight, or earth-pull, of a body changes when 
it is taken to a different latitude, or is elevated above, or 
lowered beneath, the surface of the earth (§ 68). It is 
therefore obvious that the weight of a body may change 
while the quantity of matter in it, i.e. its mass, remains 
the same. 

11. Processes of Weighing. — Since equal masses are at- 
tracted equally by the earth at any given place, weighing 



10 



A HIGH SCHOOL COURSE IN PHYSICS 



ilc 



Fig. 6. — a Dyna- 
mometer or Spring 
Balance. 




offers one of the most convenient and accurate means for 
comparing masses; thus, to make one mass equal to 
another, we have only to adjust the 
quantity of each until they stretch the 
spring of a dynamometer, Fig. 6, equally; 
or, as we say, " weigh alike " when 
placed on any weighing device. The 
usual process of determining the mass 
of a body 
consists in 
placing it 
upon one 
pan of a 
beam balance, Fig. 7, and 
known masses, called 
"weights," upon the other pan 
until the two balance. The 
sum of the known masses used 
gives the mass of the body. Such known masses, ranging 
from one milligram up to several hundred grams, consti- 
tute a so-called "set of weights." 

12. Density. — Everyday experience teaches us that 
bodies may have the same size and yet differ greatly in 
weight. A bar of iron, for example, is much heavier 
than a bar of wood of the same dimensions, because its 
mass is much greater. Tlie substances are said to differ 
in density. The density of a substance is measured by the 
number of units of mass contained iji a unit of volume. 
Thus in the C. G. S. system it is expressed as the number 
of grams per cubic centimeter ; in the common, or foot- 
pound-second (F. P. S.), system, by the number of pounds 
per cubic foot. For example, the densit}^ of lead is 11.36 
grams per cubic centimeter (abbreviated 11.36 g./cm.^). 



Fig. 7. — A Beam Balance. 



INTRODUCTION 11 



EXERCISES 

1. Find the density of a liquid of which a liter has a mass of 850 g. 

2. If the volume of a piece of glass whose mass is 10 g. is 3.9 cu. 
cm., what is the density of the glass? 

3. The density of mercury is 13.6 g./cm.^ Calculate the mass of 
mercury that can be contained in a vessel whose capacity is 30 cm.^ 

4. If mercury is 90 ct. per pound, what will half a liter cost? 

5. A vessel will hold 500 g. of mercury. How many grams of water 
will be required to fill it ? 

6. The diameter of a steel sphere is 4 cm. If the density of steel 
is 7.8 g./cm.^, what is the mass of the sphere? Volume of a sphere 

13. States of Matter. — Matter admits of being sepa- 
rated into three classes according to its ability to preserve 
(1) its shape and volume, (2) its volume only, or 
(3) neither its shape nor volume. A body that retains 
both its shape and volume is called a solid; one tliat re- 
tains its volume only and shapes itself to the vessel con- 
taining it, a liquid; while one that occupies completely 
any vessel in which it is placed, retaining neither form 
nor size, is called a gas. Ice, water, and steam are exam- 
ples of the same substance in the three states. 

14. Time. — All systems of measurement of time, used 
in Physics, employ the interval called the mean solar second 
as the unit. It is gQj-Q-Q of a mean solar day, the average 
length of time intervening between two successive transits 
of the sun's center across a meridian. 

SUMMARY 

1. Physics is the science of phenomena (§ 1). 

2. Matter is always recognized by its properties of ex- 
pansion, indestructibility, and weight. Different kinds of 
matter — gold, water, air, etc., are recognized by their 
properties; i.e. hardness, fluidity, etc. (§ 3). 



12 A HIGH SCHOOL COURSE IN PHYSICS 

3. The quantity of matter in a body is called its mass. 

4. There are two well-known systems of measurement 
for the quantities of length, area, volume, time, etc., viz. 
the English system and the metric system (§4). 

5. The units of the metric system are the centimeter, 
gram, and second. This system is the more generally 
used for scientific work. It is called the centimeter- 
gram-second (C. G. S.) system (§ 5). 

6. Units of the English system are the foot, pound, and 
second. This system is therefore called the foot-pound- 
second (F. P. S.) system (§ 12). 

7. The terms mass and weight have not the same 
meaning. The weight of a body refers to the downward 
pull of the earth upon the body. The mass of a body 
may remain constant, while the weight varies with the 
latitude, the altitude above the earth's surface, and the 
depth to which it may be lowered into the earth (§ 10). 

8. Masses are measured and compared by the process 
of weighing (§ 11). 

9. The density of a substance is measured by the num- 
ber of units of mass contained in a unit of volume (§ 12). 

10. Matter may be divided into three classes according 
to its abilit}^ to preserve (1) its shape and volume, (2) its 
volume only, or (3) neither its shape nor volume. Thus 
bodies are classed as solids, liquids, and gases (§ 13). 



CHAPTER II 
MOTION, VELOCITY, AND ACCELERATION 

1. UNIFORM MOTIONS AND VELOCITIES 

15. Motion and Rest Relative Terms. — The position of 
a body at any instant is defined by its direction and dis- 
tance from some other body which is usually conceived as 
being fixed, or at rest. Motion is a continuous change in 
the position of a body. It is customary to think of the 
earth as the fixed body when we speak of the motion of a 
train, a bird, a cloud, etc. Again, a passenger sitting in 
a moving railway coach is in the condition of rest with 
respect to the train, while with respect to the earth the 
same person is in rapid motion. Even the earth, as we 
know, is not at rest; it not only rotates on its axis, but 
travels with great speed in its orbit around the sun. 
Hence a body at rest with respect to the earth is not actu- 
ally at rest, nor is its motion with respect to the earth 
the actual motion. However, when no statement is made 
to the contrary, the earth is regarded as the body to which 
the motion of an object is referred. 

16. Path of a Moving Body. — The line described by a 
small moving body is called its path. When the path 
described is a straight line, the motion is rectilinear ; when 
curved, the motion is curvilinear. Let us first consider 
cases of rectilinear motion. 

17. Uniform Rectilinear Motion. — If a moving body de- 
scribes equal portions of its path in equal intervals of time^ 
no matter how small the intervals may be., its motion is uni- 
form. In other words, uniform motion is the motion of a 

13 



14 A HIGH SCHOOL COURSE IN PHYSICS 

body when the distance passed over is proportional to the 
time occupied. In the case of uniform rectilinear motion 
the velocity y or rate of motion, of the moving body is con- 
stant in both magnitude and direction, and is measured by 
the distance which the body travels per second, per minute, 
per hour, etc. ; for example, 10 centimeters per second 
(abbreviated 10 cm. /sec), 25 miles per hour, etc. 

18. Equation of Uniform Motion. — From § 17 it can easily 
be seen that the entire distance passed over hy a body having 
uniform motion can he found hy multiplying the velocity hy 
the time. Thus, if the velocity is 20 cm. /sec, the distance 
passed over in five seconds is 5 x 20, or 100, centimeters. 
This relation between the distance c?, the velocity v, and 
the time t is conveniently expressed by the equation 

d=vt. (l) 

19. Representation of a Motion. — In describing com- 
pletely the rectilinear motion of a body, the following 
characteristics must be given: («) the startitig point, 
(5) the direction, and (c) the distance traveled. It will 
be observed that a straight line, since it has origin, direc- 
tion, and length, is capable of representing the three char- 
acteristics of rectilinear motion. Hence the line AB, 
Fig. 8, drawn from the point A a distance of 4 centimeters 

to the right may be used to 

d , § represent the three qualities 

Fig. 8.— Representation of a of the motion of a body 4 
Rectilinear Motion. ^-^^^ -^ ^^ easterly direction 

from the place represented by the point A. By letting a 
centimeter represent 5 miles the same line will represent 
the characteristics of the rectilinear motion of a body over 
a distance of 20 miles. Thus any convenient scale may 
be used, but the same scale should, of course, be used 
throughout a given j)roblem. 



MOTION, VELOCITY, AND ACCELERATION 15 

20. Average Velocity. — Absolute uniform motion is of 
very rare occurrence, except during exceedingly small in- 
tervals of time. For instance, a train on leaving a station 
starts slowly and gains in speed until it acquires the veloc- 
ity with which it can follow schedule time. It would be 
practically impossible for the engineer so to regulate the 
throttle as to maintain an absolutely constant velocity, in- 
asmuch as the resistance due to the track, curves, wind, 
etc., would vary from time to time. Finally the throttle 
is closed, the brakes applied, and the train comes gradually 
to rest. Although the velocity has changed greatly, the 
train has passed over a certain distance in a definite time; 
let us say 30 miles in 40 minutes. The train has moved 
just as far as it would have traveled with a uniform 
velocity found by dividing the total distance by the time 
consumed, i.e. |^ of a mile per minute. This is called the 
average velocity of tlie train for the time under considera- 
tion. From this explanation it is obvious that equation 
(l) will hold for motions that are not uniform, provided 
V is the average velocity for the time t. 

21. Velocity at Any Instant. — If we examine the motion 
of all the objects with which we are familiar, we shall find, 
as in the case of the train in §20, that the velocity in 
almost every instance is either increasing or decreasing. 
Hence velocity cannot be defined accurately as the distance 
over which a body moves in a unit of time. We must, 
therefore, consider the velocity of a body at a given in- 
stant, i.e. at some stated time. The velocity of a body at 
a given instant is the distance it would move in a second if at 
that instant its motion ivere to become uniform. 

22. Representation of a Velocity. — A velocity has the 
characteristics of magnitude (i.e. speed) and direction. 
Therefore, a straight line of a definite length may conven- 
iently be used to represent the velocity of a body at a 



16 A HIGH SCHOOL COURSE IN PHYSICS 

given instant. 'Let the velocity of a body be 10 miles per 
hour north. A straight line, as AB, Fig. 9, drawn from 
^ to 5 in an upward direction and having a 
length of 10 units, will completely represent 
the given velocity. As in the case of mo- 
tions (§19), any convenient length may be 
selected to represent a unit of velocity, but 
the same length should be used throughout 
any given discussion. Any quantity, as 
^ velocity, having direction as well as magni- 

FiG. 9. — Repre- tude, is called a vector quantity, and the line 

sentation of . . 

a Velocity. representing it, a vector. 

EXERCISES 

1. A velocity of 60 mi. per hour is how many feet per second? 

2. Express a velocity of 10 m. per minute in centimeters per 
second. Draw the vector that represents the velocity when the direc- 
tion is eastward. 

3. A train travels 100 mi. in two and one half hours. Calculate 
the average velocity in feet per second. 

4. The speed of an electric car averages 20 ft. per second. How 
tar will it travel in three hours ? 

5. A train whose length is 440 yd. has a velocity of 45 mi. per 
hour. How long will it take the train to pass completely over a 
bridge 100 ft. long? Ans. 21.51 sec. 

6. A wheel 50 cm. in diameter revolves 600 times per minute. 
Express the speed of a point on its rim in centimeters per second. 

uins. 1570.8 cm. /sec. 

7. Assuming that the radius of the earth is 4000 mi. and that it 
^evolves on its axis once in exactly 24 hr., ascertain the speed of a 
point at the equator. Ans. 1047.2 mi./hr. 

2. UNIFORMLY ACCELERATED MOTION 

23. Acceleration. — We have spoken of the manner in 
which a train leaves a station. Starting from rest and 
gradually gaining in speed, it finally attains the desired 
velocity. Let us suppose that after the train has been 



MOTION, VELOCITY, AND ACCELERATION 17 

moving one second its velocity is 20 cm. /sec. ; at the end 
of two seconds from the instant of starting, 40 cm. /sec; 
at the end of three seconds, 60 cm. /sec, etc While the 
train continues to move in this manner, the velocity is 
increasing the same amount during each second, viz., 20 
cm. /sec This quantity, the rate at which the velocity 
changes with the time^ is called the acceleration of the train. 
Since the change in velocity per second is measured in 
centimeters per second, the acceleration of the train is 
conveniently expressed thus: 20 cm./sec^, and read "20 
centimeters per second per second." The acceleration of 
a body is positive or negative according as the velocity 
increases or decreases. The acceleration of a falling body, 
for example, is positive ; that of a body thrown upward, 
negative. 

24. Uniformly Accelerated Motion. — If the rate at ivhich 
the velocity of a moving body changes with the time he con- 
stant^ — i.e. if the acceleration remain uniform^ — the motion 
is called uniformly accelerated motion. The motion of the 
train considered in § 28 is of this type. There occur in 
nature many cases in which the condition that defines 
uniformly accelerated motion is very nearly fulfilled ; e.g. 
falling bodies, bodies thrown upward, bodies moving freely 
along inclined planes, etc. 

25. Velocity Acquired and Distance Traversed. — The 
velocity acquired in a given number of seconds by a body 
having uniformly accelerated motion is found in a manner 
which the following example clearly illustrates : 

A body starts from rest and gains in speed 4 cm. /sec in 
each second of time. 

At the end of one second its velocity is 4 cm. per second. 

At the end of two seconds its velocity is 8 cm. per second. 

At the end of three seconds its velocity is 12 cm. per second. 

At the end of t seconds its velocity is 4 ^ cm. per second. 
3 



18 A HIGH SCHOOL COURSE IN PHYSICS 

It is plain, therefore, that the velocity v at the end of 
any number of seconds t is found by multiplying the 
acceleration a by the time ; or 

V = at. (2) 

Again, since the initial velocity for a given interval of 
time is 0, and the final velocity is v, and since the gain in 
velocity is uniform, the mean velocity for the interval is 
(0 + v)-f- 2. If the acceleration is 4 cm. /sec. ^ as before, 

the average velocity for the first second is cm. per 

second ; 

0-4-8 
the average velocity for the first two seconds is — — — cm. 

per second ; 

the averag-e velocitv for the first t seconds is -~^ cm. 

^ " 2 

per second; and if the acceleration is a cm. /sec. 2, the av- 

T ., f. . -, . -{- at , at 

erage velocity tor t seconds is — ' cm. /sec, or — 

Now since the distance passed over is found by mul- 
tiplying the average velocity by the time (§ 20), the 
distance that the body moves in t seconds when the accel- 
eration is 4 cm./sec.2 is — ■ x ^, or ^(4^^^ cm.; but if the 

acceleration is «, we have for the distance, — x ^, or J a^. 

Hence d = iat2. (3) 

Example. — A car starts to move down an incline that is just steep 
enough to cause it to gain in velocity 50 cm. /sec. during each second 
of time. Find the velocity and the distance traversed at the end of 
the fifth second, and the distance that the car moves during the fifth 
second. 

Solution. — The initial velocity is 0. The final velocity is the 
product of the gain per second and the number of seconds. Hence 
t' = 50 X 5, or 250 cm./sec. 



MOTION, VELOCITY, AND ACCELERATION 



19 



The total distance traversed is the product of the average velocity 
multiplied by the number of seconds. The average velocity for 5 



seconds is 
. 250 



+ 250 



and the number of seconds is 5. Hence 



X 5, or 625 cm. The same result could be found by sub- 

stituting the values of a and t in equation (3). The analysis of a 
problem, however, is far more valuable than the mere substitution of 
numbers in a formula. 

The distance traversed during the fifth second is evidently the 
difference between the distance traversed in 5 seconds and that in 



4 seconds. Now in 4 seconds the car moves 



+ 4 X 50 



X 4, or 400 cm. 



Hence during the fifth second the car moves 625 — 400, or 225 cm. 

We thus observe that if any two of the four quantities 
used in the discussion be given, the others can be calcu- 
lated by the help of equations (2) and (3). 

The relation of time and distance shown by equation (3) 
may be tested experimentally as follows : 

Let a grooved board AB, Fig. 10, about 15 feet long be supported 
with one end elevated about 18 inches. The groove can easily be 
formed by nailing a strip of wood about 2 inches wide to the side of 




Fig. 10. — Verifying the Relation between the Distance Traversed and the Time 
in Uniformly Accelerated Motion. 

a wider piece, forming a cross section as shown in X Sufficient sup- 
ports should be used to keep the groove straight. Arrange a seconds 
pendulum (§ 84) so as to make and break an electrical contact at the 
center of its path, and connect a telegraph sounder and a cell of 
battery in the circuit with the pendulum. The sounder should give 



20 A HIGH SCHOOL COURSE IN PHYSICS 

a loud click at the end of every second. Release a marble at A pre- 
cisely at the instant the sounder clicks, and place a block C at such 
a point on the incline that the click of the marble against C coincides 
with the click that marks the end of the third second. This point 
will have to be found by trial, and should be verified by two or three 
tests. The length A C gives the distance passed over by the marble 
in three seconds. Let the process be repeated for two seconds and 
one second. The distances found should be proportional to the 
squares of the times, as shown by equation (3) ; i.e. as 1 : -i : 9. 

EXERCISES 

1. Solve both equations (2) and (3) for the acceleration a and the 
time t. 

2. Combine equations (2) and (3) so as to express the velocity in 
terms of the acceleration a and the distance d. Express also the dis- 
tance in terms of velocity and acceleration. 

3. Letting a line 1 cm. long express the acceleration a, represent 
the velocity at the end of each of the first four seconds. Represent 
also the corresponding distances. 

Suggestion. — Equation (2) gives the length representing the 
velocity, and equation (3), the distance. 

4. A train leaving a station has a constant acceleration of 
0.4 m./sec.2. What will be its velocity at the end of the tenth 
second? At the end of 15 seconds? 

5. If the acceleration of an electric car is uniform and 2 ft./sec.^, 
in how many seconds will it accumulate a velocity of 25 ft. per 
second ? 

6. How far will the car in Exer. 5 move during the first 10 seconds ? 
What will be its average velocity during this interval of time? 

7. The acceleration of a car is 5 m./sec.^. What velocity will it 
acquire in going 100 m.? Ans. 31.62 m./sec.^. 

8. A body has uniformly accelerated motion. What is its accel- 
eration if it passes over 300 cm. in 20 seconds? ^Lns. 1.5 cm./sec.^. 

9. A bicycle starts from rest at the top of a hill 150 ft. long and 
has a uniform acceleration of 1 ft. per second. What will be its 
velocity at the foot of the hill? Ans. 17.32 ft./sec. 

10. A car was moving at the rate of 30 mi. per hour when the 
brakes were applied. What was the rate of retardation if the car 
came to rest in 10 seconds, the decrease in velocity being uniform ? 

Ans. 4.4 ft./sec.2. 



MOTION, VELOCITY, AND ACCELERATION 21 

11. A bicycle rider moving at the rate of 15 mi. per hour applies 
the brake which brings him to rest in moving 121 ft. Assuming 
that the velocity decreases uniformly, find the acceleration. 

Ans. — 2 ft. /sec. 2. 

12. A fly wheel is set in motion with a uniform acceleration of 
two revolutions per second per second. If the diameter of the wheel 
is 50 cm., what is the linear acceleration of a point on its rim ? 

Am. 314.16 cm./sec.2. 

13. What is the velocity of a body having uniformly accelerated 
motion at the beginning of the ^th second? What is the average 
velocity during the ^th second? Show that the distance passed over 
during the ^th second is | a (2 f — 1). 

14. Apply the formula developed in Exer. 13 to the conditions 
given in Exer. 4, and calculate the distance passed over by the train 
during the fifth and the tenth second. Ans. 1.8 m. and 3.8 m. 

3. SIMULTANEOUS MOTIONS AND VELOCITIES 

26. Composition of Motions. — The actual displacement 
of a body is often due to two or more causes acting to- 
gether. For example, a ball rolled along the deck of a 
vessel has a displacement that is the result of combining 
the displacement of the boat with that given the ball by the 
hand. Hence, to find the displacement of the ball with 
respect to the earth, we must take into account all the 
separate motions that enter into the case. It will be ob- 
served that different cases will arise depending on the rela- 
tion of the magnitudes and directions of the displacements 
to be compounded. The individual motions effecting the 
displacement are called the components, and the motion 
due to the united action of the components, the resultant. 
The process of finding the resultant from the components 
is called the composition of motions. 

27. Compounding Motions in a Straight Line. — Let a 
ball be rolled along the deck of a vessel toward the bow. 
While the ball moves over a distance of 20 feet along the 
deck, the boat moves forward 30 feet. Since the com- 



22 A HIGH SCHOOL COURSE IN PHYSICS 

ponent motions are in the same direction, it is plain that 
the ball actually moves a distance of 50 feet in the direction 
the vessel is going. Hence the resultant is equal to the 
sum of the components. Again, imagine that the ball is 
rolled toward the stern, a distance of 20 feet, while the 
boat is moving forward 30 feet. In this case we can see 
that the ball will be carried forward by the vessel 10 feet 
farther than the distance which it rolls backward. Hence 
the resultant is 10 feet, and in the direction of the motion 
of the vessel, because that is the larger component. A 
rule for compounding motions in the same straight line 
may be stated as follows: 

The resultant of two component motions in the same straight 
line is equal to their sum when the directions are the same^ 
and to th'eir difference when the directions are opposite. In 
the latter case the resultant is in the direction of the greater 
component. 

28. Compounding Velocities in a Straight Line. — If 

the velocities of the boat and the ball considered in § 27 
are respectively 15 and 10 feet per second, and if the 
directions are the same, it is clear that the actual velocity 
of the ball will be the sum of the velocity of the boat and 
the velocity given to the ball by the hand, or 25 feet per 
second; and if the directions are opposite, the actual 
velocity of the ball will be equal to the difference of 
the velocities, or 5 feet per second. In the latter case 
the resultant velocity is in the direction of the boat's 
motion, since that is the greater of the two components. 

29. Compounding Motions at an Angle. — Let a man 
starting from the point A, Fig. 11, row a boat perpen- 
dicular at all times to the current of a river 40 rods 
in width. If there were no current, the boat would land 



MOTION, VELOCITY, AND ACCELERATION 



23 



at B. But, while crossing, the current carries the boat 
down the stream a distance AC^ which we may call 30 
rods. The boat therefore lands at 2>, having taken the 
path AB. Since AD is in 
this case the diagonal of a 
rectangle whose sides are 
40 and 30 units, represent- 
ing distances measured in 
rods, its length, which is 
50 units, represents a dis- 
tance of 50 rods, the re- 
sultant motion of the boat. 
If the angle between the 
components is not a right 
angle, a boat starting from 
E takes a path EH^ the di- 
agonal of an oblique paral- 
lelogram EFHa. 




Fig. 11. — Resultant of Two Motions 
at an Angle. 



R=£ 



m E 
1 



A thin piece of wood E, Fig. 
12, is arranged to slide smoothly 
along the edge of a drawing board. At m and n wire nails are driven 

a short distance into the wood and a 
third into the board at o. Loop one 
end of a piece of thread around m, 
pass it over n, and attach the other 
end to a small weight at A. If the 
board is now placed in a vertical po- 
sition and the slide moved from E to 
E', the weight undergoes a displace- 
ment AB. If the loop is transferred 
to nail 0, on the stationary board, a 
movement of the slide from E to E' 
gives the weight two simultaneous 
displacements represented by AB and BD, causing it to follow the 
diagonal path AD. If the operations are repeated after tilting the 
board in its plane, it will be seen that the weight follows the diagonal 



r -AD 

/\ 
/ I 



Fig. 12. — Apparatus for Show- 
ing the Compounding of Two 
Motions. 



24 



A HIGH SCHOOL COURSE IN PHYSICS 



of an oblique parallelogram as the result of the two component dis- 
placements. 

The facts shown by this experiment may be stated as follows : 

The resultant of two component uniform motions not in the 
same straight line is represented hy the diagonal of a paral- 
lelogram whose adjacent sides represent the two component 
motions. 

30. Compounding Velocities at an Angle. — Velocities 
may be compounded in the same manner as motions. For 
example, if the velocity with which an oarsman rows his 
boat is represented in magnitude and direction by the line 
EF in Fig. 11, and the velocity of the stream by the line 
_E'(r, the actual velocity of the boat is represented by the 
line EH, 

It should be observed that in cases where the angle between 
the two components is a ri(jht angle, the resultant is the square root 
of the sum of the squares of the components. In other cases the 
parallelogram should be constructed accurately and the diagonal care- 
fully measured. 

31. Compounding Several Motions. — When the actual 
motion of a body is due to the united action of more than 
two components, the final resultant is found by first deter- 
mining the resultant 
of any two of them; 
and this resultant is 
then compounded with 
a third component, and 
so on until each com- 
ponent has been used. 

Let AB, AC, and AD, 

Fig. 13, be three component 

-N^/ motions imparted simul- 

FiG~lI-DeTe'-mTn7th'n7rt"^^^^ of ^aneously to a body at A. 

Three Component Motions. The resultant of any two, 



^v^ 



'^v 



^v. 



■=*.F 




MOTION, VELOCITY, AND ACCELERATION 25 

e.g. AB and AD, is found in the manner described in § 29, giving the 
resultant AE. AE is now treated as a component and compounded 
with the third component A C. In this construction A CFE is the par- 
allelogram of which ^i^ is the diagonal. AF \s, the resultant of the 
three given components. 

32. Resolution of Motions and Velocities. — The meaning 
of this process, which is the reverse of composition, is most 
readily understood after considering a particular case. 
For example, let it be required to find the easterly velocity 
of a vessel sailing with a velocity of 15 miles per hour in 
a direction east by 30° south. ^ component jj 

Let AB, Fig. 14, represent the 
given velocity of the vessel, making 
the angle BA C equal to 30°. If the 
lines BD and BC are now drawn 
parallel to AC and AD respectively, 
the line AC represents the com- « 
ponent velocity of the vessel in an Fig. M.- Resolution of a Velocity. 

easterly direction. AD is the southerly component. 

EXERCISES 

1. A train approaches Chicago with a velocity of 30 km. per hour, 
while a brakeman runs along the tops of the cars toward the rear at 
the rate of 5 km. per hour. How rapidly is the brakeman approach- 
ing Chicago ? 

2. A boy is paddling a canoe along a river in the direction of the 
current, which has a velocity of 4 mi. per hour ; if there were no cur- 
rent, the canoe would move 3.5 mi. per hour. How fast is the canoe 
moving? 

3. Suppose the boy in Exer. 2 should double his effort and paddle 
upstream. How long would it take him to go 10 mi. ? 

4. A ship is moving east at the rate of 15 mi. per hour. If a 
person walks directly across the deck at the rate of 4 mi. per hour, 
with what velocity will he actually move ? 

5. A boat is rowed with a velocity of 4 mi. per hour, perpendicu- 
lar to the current of a stream flowing 5 mi. per hour. Determine the 
direction of the motion and the velocity of the boat. 

6. A ship headed due east under a power that can move it 12 mi. 



26 A HIGH SCHOOL COURSE IN PHYSICS 

per hour enters an ocean current whose velocity is 4 mi, per hour south. 
If a person on deck walks northeast with a velocity of 3 mi. per hour, 
what is his actual velocity? Ans. 14.3 mi./hr. 

7. A balloon is driven in a direction east by 30° north. How 
rapidly is it drifting north if its velocity is 20 mi. per hour ? 

Ans. 10 mi./hr. 

8. A body moves down an inclined plane 5 m. in length. If the 
angle between the incline and a horizontal plane is 60°, what are the 
horizontal and vertical components of its motion ? 

9. How rapidly is a bird approaching the equator when flying 
due southeast at the rate of 20 mi. per hour ? 

SUMMARY 

1. Motion is a continuous change in the position of a 
body. Motion and rest are relative terms. When the 
motion of terrestrial bodies is under consideration, the 
earth is usually regarded as being at rest (§ 15). 

2. The motion of a body is rectilinear or curvilifiear 
according as the path described by the body is a straight 
or a curved line (§ 16). 

3. When a body moves over equal spaces in equal 
periods of time, no matter how small the period may be, 
the motion is said to be uniform. When the motion of a 
body is uniform, the distance passed over is proportional 
to the time (§ 17). 

4. The equation of uniform motion in d = vt (§ 18). 

5. The characteristics of the rectilinear motion of a 
body are its starting pointy the direction of the motion, and 
its displacement. These three qualities may be represented 
by a straight line (§ 19). 

6. The average velocity of a body is found by dividing 
the space passed over by the time consumed (§ 20). 

7. The velocity of a body at any instant is measured 
by the distance it would move in a second if at that instant 
its motion were to become uniform (§ 21). 



MOTION, VELOCITY, AND ACCELERATION 27 

8. The characteristics of a velocity are its magnitude 
(or speed) and direction. The qualities may be repre- 
sented by the length and direction of a straight line. A 
line used in this manner is called a vector (§ 22). 

9. The rate at which velocity changes with the time 
is called acceleration (§ 23). 

10. A body has uniformly accelerated motion when its 
velocity changes at a uniform rate. The velocity acquired 
in a given time by a body starting from rest may be found 
from the equation v = at. The distance passed over is 
given by the equation d = ^at'^ (§ 24). 

11. The simultaneous individual motions (or velocities) 
of a body are called the components of its motion (or ve- 
locity), and the motion (or velocity) due to the united 
action of the components is called the resultant. The pro- 
cess of finding the resultant from the components is called 
the composition of motions (or velocities) (§§ 26 and 28). 

12. The resultant of two simultaneous motions (^or veloci- 
ties) along the same straight line is their sum when the 
directions are the same, and their difference when they 
are opposite (§§ 27 and 28). 

13. The resultant of two simultaneous motions (^or veloci- 
ities) not in the same straight line is represented in magni- 
tude and direction by the diagonal of a parallelogram 
whose adjacent sides represent the two components (§§ 29 
and 30). 

14. The resultant of more than two simultaneous motions 
(or velocities') is found by compounding the resultant of 
any two of them with a third component, then this new 
one with the fourth, and so on until eacli component lias 
been used (§ 31). 

15. The process of finding the components from the 
resultant is called resolution (§ 32). 



CHAPTER III 
LAWS OF MOTION — FORCE 

1. DISCUSSION OF NEWTON'S LAWS 

33. Momentum. — It is a well-known fact that a mov- 
ing body must always have been put in motion by an effort 
on the part of some agent. The amount of this effort de- 
pends (1) on the mass of the body moved and (2) on the 
rapidity with which it is given velocity, i.e. on the acceler- 
ation. If we observe a locomotive as it starts a train, we 
readily see that the effort required is greater as the train 
is longer or more heavily loaded. Furthermore, in order 
to start the train more quickly, a greater effort is required 
and a more powerful engine. Again, after the train has 
acquired its running speed, an effort is required if the 
motion is to be destroyed and the train brought to rest. 
This, also, depends on the mass of the train and the rapid- 
ity with which its motion is reduced. 

The two quantities, mass and velocity., determine what is 
called the quantity of motion in a body, or its momentum. 
Momentum is measured hy the product of the mass and the 
velocity of a body, and is expressed algebraically as mv. 
For example, the momentum of a 10-gram rifle ball mov- 
ing with a velocity of 25,000 cm. /sec. is 10 x 25,000, or 
250,000 C. G. S. units. No name is used for the unit of 
momentum. 

34. Newton's First Law of Motion. — An inanimate body 
never puts itself in motion. Not only does a body with- 
out motion tend to remain in that condition, but on ac- 

28 



LAWS OF MOTION — FORCE 29 

count of that same tendency resists the effort of any agent 
that tries to start it. On the other hand, a body in mo- 
tion manifests a tendency to keep moving and resists any 
effort made to change its motion in any way. These 
facts may be illustrated by the following experiments : 

1. Stand a book upon end on a sheet of paper placed flat upon the 
table. Grasp the paper and try by a quick pull to give the book a 
forward motion. On account of the tendency of the book to remain 
at rest, it will be found to fall backward. Repeat the experiment, 
but move the paper very slowly at first ; and, while the book is in 
motion, let the paper suddenly stop. On account of the fact that the 
book tends to remain in motion, it will fall forward. 

2. Place a card upon the tip of a finger and lay a small coin upon 
the card directly above the finger tip. With the other hand give 
the card a sudden snap in such a manner as to drive the card from 
beneath the coin. The coin will be left upon the finger. The same 
experiment may be varied by placing a card upon the top of a bottle 
and a marble upon the card. The sudden removal of the card leaves 
the marble resting in the mouth of the bottle. 

The first to express these facts of common observation 
in the language of Physics was Sir Isaac Newton ^ (1642- 
1727), professor of mathematics at Cambridge, England. 
The statement of his First Law of Motion is as follows: 

Every body of matter continues in its state of rest or of 
uniform motion in a straight line^ except in so far as it is 
compelled by force to change that state.^ 

This is known as the Law of Inertia, the tendency of 
matter to act in the manner stated being often ascribed to 
a property of matter called inertia. In this law Newton 
has given a definition of force as that which is able to 
cause or change motion. Hence the term force is the name 

1 See portrait facing p. 30. 

2 ' ' Every body perseveres in its state of rest, or of uniform motion in 
a right line, unless it is compelled to change that state by forces impressed 
thereon." — Newton's Principia^ Motte's Translation. 



30 A HIGH SCHOOL COURSE IN PHYSICS 

given to the cause that produces acceleration^ retardation^ or 
a change in the direction of the motion of a body. 

35. Newton's Second Law of Motion. — According to 
Newton's First Law a moving body which could be en- 
tirely freed from the action of all forces would have uni- 
form motion. A stone thrown from the hand would take a 
perfectly straight couj-se, and a bullet fired upward would 
never return to the earth. The curved path described by 
the stone, however, indicates that a force is acting upon 
the body while it is moving. This force, as we know, is 
the force of gravity. 

Just as the First Law defines force^ so the Second Law 
leads to the measurement of force. This law may be stated 
as follows : 

Change of motion^ or momentum^ is proportional to the 
acting force and takes place in the direction in which the 
force acts.^ 

The proportion always existing between force and the 
rate at which it changes the momentum of a body has led 
to the adoption of a convenient C. G. S. unit of force 
called the dyne (pronounced dine'). The dyne is that 
force which., acting uniformly for one second^ imparts one 
C. Gr. iS. unit of momentum; or, a dyne would give a mass 
of one gram an acceleration of one centimeter per second 
per second. This definition implies that the body upon 
which the force acts is not at all hindered in its motion 
by external resistances, as friction, etc.; i.e. the force has 
simply to overcome the inertia of the body. 

36. Equations of Force. — The relation expressed in the 
preceding section between force, momentum, and time ad- 

1 "The alteration of motion is ever proportional to the motive force 
impressed ; and is made in the direction of the right line in which the 
force is impressed." — Newton's Principia, Motte's Translation. 




SIR ISAAC NEWTON (1642-1737) 

The name of Newton will always be associated with the subject 
of gravitation on account of the fullness with which he applies and 
discusses his famous Principle of Universal Gravitation in his book 
entitled Principia, published in 1687. The Prmcipia, which ranks as 
a mathematical classic, treats of the laws governing the motion of 
bodies under various conditions, and especially of the motion of the 
planets. This work follows close upon the achievements of Galileo 
and Kepler in astronomical discovery. Kepler had found by obser- 
vation that the planets move around the sun in elliptical paths, which 
Newton showed would be the case if between the sun and each 
planet there exists a force which decreases as the square of the 
distance increases. 

The Principia laid a firm and deep foundation for subsequent 
discoveries in the field of astronomy; it propounded and showed the 
application of a new method of mathematical investigation, the Cal- 
culus, by which alone it would retain its position at the head of 
mathematical treatises. 

Newton must also be accredited with the announcement and elu- 
cidation of the three laws of motion which bear his name and with 
numerous discoveries in Light. His book entitled Optics contains 
his discoveries and theories in this subject. 

Newton was born in Lincolnshire, England, in 1643, graduated 
from Trinity College, Cambridge, in 1665, and at once began to make 
the discoveries in mathematics and physics which have immortal- 
ized his name. He was professor of mathematics at Cambridge, 
member of Parliament, and Master of the Mint. He was knighted 
in 1705, and at his death, in 1737, was buried in Westminster Abbey. 



LAWS OF MOTION — FORCE 31 

mits of being expressed in a concise algebraic form. Let 
a mass of m grams be acted upon by a constant force of/ 
dynes for t seconds. If the velocity imparted to the mass 
by that force is v cm./sec, the momentum produced is mv 
(§ 33). The momentum imparted per second is found by 
dividing the quantity mv by the time t. We have there- 
fore, by the definition of the dyne, 

J. .. J X m(inp'rams) x v (in cm./sec.) ^,x 

f Tin dynes) = — ^^ — ^ ^^ '- (l) 

^ -^ ^ t (in seconds) ^ ^ 

From Eq. (2) § 25, we have v = at\ 
whence - is the acceleration a produced by the force /. 

Therefore, by substituting in (l), 

f (in dynes) = m (in grams) x a (in cm./sec.^). (2) 

In the English F.P.S. system the unit of force is the poundal. The 
poundal u that force lohich, acting uniformly for one second, imparts one 
F. P. S. unit of momentum. Hence, if the mass of the body upon which 
the force acts is given in pounds, the velocity in feet per second, and 
the time in seconds, substitution in equation (1) will give us the force 
in poundals. A force of one poundal is equivalent to 13,825 dynes. 

Example. — What constant force acting four seconds will give a 
body of 15 g. a velocity of 20 cm./sec. ? 

Solution. — The total momentum produced by the force in four 
seconds is 15 x 20, or 300 C. G. S. units. The momentum imparted 
to the body in one second is 300 -^ 4, or 75 units. Hence the force 
is 75 dynes. 

Another method is to substitute the given quantities directly in 
equation (1), first being assured that all are given in C. G. S. units. 
The same may be said of the F. P. S. system in obtaining the force in 
poundals when the problems deal with English units. 

37. Gravitational Units of Force. — A gram-weight (or 
a gram-force) is the downward pull exerted by the earth 
upon a one-gram mass. (See §10.) If the mass is free 
to fall, the acceleration due to gravity will be 980 
cm. /sec. 2, or 980 C. G. S. units of momentum per second. 



32 



A HIGH SCHOOL COURSE IN PHYSICS 



Hence, by Eq. (2), a gram-weight is equivalent to 980 dynes. 
Likewise, a pound-weight (or a pound-force^ is the attrac- 
tion of the earth upon a one-pound mass. The accelera- 
tion produced by this force when the body falls freely is 
32.16 ft./sec.2, and the momentum is 32.16 F. P. S. units 
per second. Therefore, one pound-weight is equivalent to 
32.16 poundals. 

Since the earth's attraction varies with the locality, as 
explained in §§10 and 69, the gram-weight and the pound- 
weight change accordingly. Hence these units are called 
gravitational units of force. On the other hand, the dyne 
and the poundal, being independent of gravity, are called 
absolute units. 

Commonly, use is made of the pound and the gram (i.e. 
pound-weight and gram-weight) as units of force. Scien- 
tific work has, however, demanded a more unvarying unit 
and has brought the dyne into extensive use in all cases 
requiring accuracy of expression. The poundal is little 
used. 

38. An Application of the Second Law. — Newton's 
Second Law implies that any force acting upon a body 
produces its own effect, whether acting alone or conjointly 
with other forces. An interesting illustration of this may 

be observed in the fol- 
lowing experiment : 



Cut notches A and B, 
Fig. 15, in two of the cor- 
ners of a piece of wood about 
2x8 inches. By means of 
a large screw attach the cen- 
ter of the block loosely to the 
edge of a table as shown, and 
place a marble in each notch. 
If the end of the block op- 
posite A is struck with a 




Fig. 15. — Illustration of Newton's Second 
Law of Motion, 



LAWS OF MOTION — FORCE 



33 



mallet, ball A will be dropped vertically, while ball B is projected hor- 
izontally. B is subject to two forces, an impulse which projects it in 
a horizontal direction and the constant force of gravity acting along 
a vertical line. The two marbles will be found to strike the floor at 
the same time. 

The experiment shows that the effect of gravity in 
bringing the balls to the floor is independent of the hori- 
zontal component of the motion; i.e. a given force (gravity) 
produces as much " change in momentum " in the vertical 
direction in one ball as in the other. After ball B leaves 
the block, its momentum in the horizontal direction suffers 
no change. 

39. Newton's Third Law of Motion. — To every action 
there is always an equal and opposite reaction; or, the mu- 
tual actions of any tivo bodies are always equal in magnitude 
and oppositely directed. 

This law may be illustrated by experiment as follows : 



Let two elastic w^ooden balls A and 
B, Fig. 16, be suspended by threads in 
such a manner that they just touch each 
other when stationary. Draw A aside 
and let it fall against B. A will be 
brought to rest by the impact, and B will 
be moved to the position B'. 



<n1 



\ 



A £ 

Fig. 16. — Action and Reac- 
tion are Equal and Oppo- 
site. 



In this experiment two results 

are apparent: first, ball^ is acted 

upon by a force sufficient to carry 

it to the position B' ., and, second, ball A loses an equal 

amount of momentum in that it is brought to rest. In 

the impact occurs a mutual effect — a force exerted by 

A toward the right upon B^ and an equal and oppositely 

directed force from B upon A. This process goes on in 

every case in which force enters. A pressure of the hand 

against the table is opposed by an equal pressure of the table 

against the hand. When a person leaps forward from a 
4 



34 A HIGH SCHOOL COURSE IN PHYSICS 

boat, the boat is pushed in the opposite direction. When 
a gun is fired, the mutual effect is to give the bullet and 
the gun equal momenta; or, in other words, the mass of 
the bullet multiplied by its velocity equals the mass of 
the gun multiplied by its velocity. The velocity of the 
gun's recoil, or "kick," is small because the mass of the 
gun is many times greater than that of the bullet. 

The question often arises : " Does the earth rise to meet 
the falling apple ? " In the light of the Third Law of 
Motion, we must admit that the action of the earth which 
draws the apple down is accompanied by a reaction which 
is operative for the same length of time upon the earth in 
an upward direction. Hence the momentum given the 
apple equals that given the earth. On account of the 
enormous mass of the earth, however, the distance through 
which it rises to meet the apple is infinitesimally small. 

EXERCISES 

1. Why does a person standing in a car tend to fall backward 
when the car starts, and forward when it stops? 

2. Why does a bullet continue to move after leav- 
ing a rifle? 

3. A weight W, Fig. 17, is attached by a cord B 
to some fixed object. A quick downward pull on a 
similar cord A will break the cord below W, but a 
steady pull will break cord B. Explain, 

4. A blast of fine sand driven against glass soon 
cuts away its smooth surface. AVhy ? 

5. If a rifle ball is thrown against a board placed 
on edge, it will knock it down ; but when fired from 
a gun, it will pass through the board and leave it 
standing. Why ? 

6. Explain why the head of a hammer or mallet 
Fig 17 ^^^ ^^ driven on by simply striking the end of the 

handle. 
7. Why can an athlete make a longer "running jump" than a 
" standing" one? 




LAWS OF MOTION — FORCE 35 

8. Will a stone dropped from a moving train fall in a straight 
line? 

9. Why do moving railway coaches " telescope " in a collision? 

10. Explain how heavy fly wheels serve to steady the motion of 
machinery, as in the case of the sewing machine. 

11. A 4-gram rifle ball leaves a gun with a speed of 20,000 cm. per 
second. Compute its momentum. 

12. Which has the larger momentum, a man weighing 150 lb., 
walking 10 ft. per second, or a boy weighing 60 lb. and running 25 ft. 
per second ? 

13. Which has the greater momentum, a man weighing 160 lb. 
in a railway coach moving 30 mi. per hour, or a 2-ton stone moving 
3 ft. per second? Express the difference in F. P. S. units. 

14. What force acting for 10 seconds upon a mass of 200 g. will 
produce a velocity of 5 cm. /sec? Express the change of momentum 
per second in C. G. S. units. 

15. A body whose mass is 20 g. is given an acceleration of 
45 cm./sec.^. What is the required force? 

16. What acceleration will be given to a mass of 25 g. by a 
constant force of 500 dynes ? Over what distance will the body move 
in 5 seconds if the force continues to act ? 

17. If the force given in Exer. 16 ceases to act at the end of the 5th 
second, how far will the body move during the next 5 seconds ? 

Suggestion. — Find the velocity imparted in the first 5 seconds 
and apply the First Law of Motion. 

18. An inelastic ball of clay whose mass is 200 g. has a velocity 
of 25 cm. /sec. when it collides with a similar ball at rest whose mass 
is 50 g. Find the velocity after collision. 

Suggestion. — After impact the two masses move on as one mass 
with the momentum of the first before collision. 

19. A projectile weighing 100 lb. is fired with a velocity of 1200 ft. 
per second from a gun weighing 8 T. Find the velocity with which 
the gun starts to move backward. 

2. CONCURRENT FORCES 

40. Representation of Forces. — The three characteristics 
of a force, its point of application, direction, and magni- 
tude, can, as we have seen, be represented by a straight 
line. One end of the line shows the point of application, 



36 



A HIGH SCHOOL COURSE IN PHYSICS 




15 Dynes 



->B 



Fig. 18. 



(2) 

The Representation of Forces by Means 
of Straight Lines. 



the length of the line shows the magnitude of the force, 
and the direction in which the line is drawn shows the 
direction in which the force acts. For example, (1), 

Fig. 18, represents 
a force of 10 dynes 
acting northeast 
from the point A. 
The unit of force 
maybe represented 
by any convenient 
length, but the 
same scale should, 
of course, be used throughout a given problem. 

In a similar manner, (2), Fig. 18, shows that two 
concurring forces, AB and AC, representing respectively 
15 dynes east and 8 dynes north, act upon the point A. 
The scale adopted in this case is 2 millimeters to the 
dyne. 

41. Composition of Forces. — When a body is acted 
upon by two forces at the same time, it is easy to imagine 
a single force that might be substituted for them and 
would have the same effect. This single force is the 
resultant of the two forces, which are the components. 
The process of finding the resultant of ttvo or more component 

forces is called the composition of forces. Forces are com- 
pounded in the same manner as motions and velocities 
(§§ 27-31). 

42. Forces Acting in a Straight Line. — When two 
forces act upon a body in the same line and in the same 
direction, it is clear that the resultant is the sum of the 
two components. For example, if a weight is to be lifted 
by two men pulling upward on a rope attached to it, and 
if one man pulls with a force of 50 pounds while the other 
pulls with a force of 75 pounds, the two forces result in a 



LAWS OF MOTION — FORCE 37 

single pull of 125 pounds. Hence the resultant is 125 
pounds and is directed upward. 

When the two forces act in opposite directions along 
the same line, the resultant is the difference of the two 
components. Thus, if one man pulls upward on a weight 
with a force of 75 pounds while the other pulls downward 
with a force of 50 pounds, the lifting effect is the same as 
a single force of 75 — 50, or 25 pounds. The action of the 
resultant is plainly in the direction of the greater of the 
two components. A special case of opposite forces is that 
in which the sum of the components acting in one direction 
is equal to the sum of those acting along the same straight 
line in the opposite direction. In this case no motion can 
result from the joint action of all the forces. In other 
words, the resultant is zero. The body upon which such 
forces act is said to be in equilibrium. 

43. Forces Acting at an Angle. — If AB and AG, (2), 
Fig. 18, represent forces of unequal magnitudes, it is clear 
that the resultant will divide the angle between them, but 
will lie nearer the greater force. The actual magnitude 
and direction of the resultant are found in the same man- 
ner as in the case of the resultant of two motions (§ 29). 

The resultant oftivo concurring forces acting at an angle is 
represented hy the diagonal of a parallelogram constructed on 
the two lines representing the component forces. 

This is one of the most important laws of mechanics and 
is universally known as the Principle of the Parallelogram 
of Forces. The following experiment will illustrate the 
truth of this principle: 

Arrange two dynamometers (see Fig. 6), before the blackboard, 
as shown in Fig. 19. Let the weight W be gi-eat enough to produce a 
large but measurable tension in each of the oblique cords. Place a 
rectangular block of wood against each of the cords and trace its 
direction on the blackboard. Read the dynamometers and record the 



38 



A HIGH SCHOOL COURSE IN PHYSICS 



magnitude of each force on the corresponding line. Adopt a conven' 
lent scale and lay off each force along its line of direction, measuring 

from O. Using the ob- 
lique lines as sides, con- 
struct the parallelogram. 
Measure the diagonal OR 
and write its value in 
force units upon it. A 
comparison will show that 
the force represented by 
OR is equal to W and 
might, therefore, be sub- 
stituted for the two 
components and would 
produce the same effect. 

44. Equilibrant and 
Resultant. — The 




Fig. 19. — Principle of the Parallelogram of 
Forces Illustrated. 



force W is said to hold the component forces in equilib- 
rium^ and is therefore called the equilibrant (pronounced 
e' quill' brant). From the definitions given of resultant 
and equilibrant, we see that they are necessarily equal 
in magnitude but opposite in direction. 



EXERCISES 

1. Represent by a diagram the resultant of two forces of 15 dynes 
and 25 dynes acting (1) in the same direction and (2) in opposite 
directions from the same point. 

2. Find the magnitude and the direction of the resultant of two 
forces, 3 lb. acting north and 4 lb. acting west, applied at the same 
point. 

3. A ball is acted upon simultaneously by two forces, one of 10 kg. 
directed upward, the other of 25 kg. directed east along a horizontal 
line. Find the resultant in both magnitude and direction. 

4. The angle between a force of 50 dynes and one of 30 dynes is 
60^. Find the resultant and the equilibrant in both magnitude and 
direction. 

5. The angle between two equal forces of 40 lb. each is 120°. Find 
the resultant. 



LAWS OF MOTION — FORCE 39 

6. A boat is pulled by two ropes making an angle of 30°. If one 
force is 10 lb. and the other 20 lb., what is the resultant? 

7. A weight is suspended by two cords applied at the same point 
and each making an angle of 30° with a vertical line. If the tension 
in each is 25 lb., what is the weight supported ? 

Ans. 43.3 lb. 

3. MOMENTS OF FORCE — PARALLEL FORCES 

45. The Moment of a Force. — It can often be observed 
that when a mechanic wishes to loosen a nut that is diffi- 
cult to start, he uses a wrench with a long handle. For 
those that start easily, he uses a short-handled wrench. 
The results prove that the effectiveness of a force in pro- 
ducing rotation against a resistance is greater as the ap- 
plied force is farther from the point about which rotation 
takes place. 

The effectiveness of a force in producing a rotation is called 
the moment of the force. The moment of a force depends 
upon two quantities: (1) the magnitude of the force and 
(2) the perpendicular distarice from the point about which 
the rotation takes place to the line represeiiting the direction 
of the force. The moment is measured by the product of 
these two factors. The following experiment will make 
the matter clear: 

Fasten one end of a light wooden bar to the table top by means of 
a nail at 0, Fig. 20. Let a force, which 
may be measured by a dynamometer (see 
Fig. 6), pull upon the bar at A, and an- 
other at B, as shown. Measure both forces 
and the distances AO and BO. The prod- 
uct of the force applied at A multiplied 
by the distance A will be found equal to 
the product of the other force multiplied 
by the distance BO. 

It is clear from this experiment 

.^ ^ J, 7 , , , Fig. 20. — The Equality of 

that ajorce that tends to turn a body Moments illustrated 



40 



A HIGH SCHOOL COURSE IN PHYSICS 



to the right can he balanced by another of the same moment 
that tends to produce rotation to the left. 

46. Parallel Forces. — Objects are frequently supported 
by two or more upward forces acting at different points, 
thus forming a system of parallel forces. For example, 
two men may support a heavy beam or carry a loaded 
bucket on a bar between them. A bridge is supported by 
the upward pressures of the piers at the ends. The prin- 
ciple of moments given in the preceding section is of 
service in determining the resultant of such forces as 
the following experiment illustrates: 

Select a bar of wood 4 or 5 ft. in length, of uniform width and 
uniform thickness. A pine board about 4 in. in width is convenient. 

Place hooks at several points 
along one edge, as shown in 
Fig. 21, but place one hook 
so that the bar will balance 
well when hung from that 
point. Call this point C. 
Suspend the bar from a dy- 
namometer at C and ascer- 
tain the weight of it. This 
will be the value of the re- 
sultant in every case. Now release the bar at C and attach dynamom- 
eters at two points, say A and B, and ascertain the forces required 
to support the bar. Designating these forces by F and F', it will be 
found that in every case the sum of the two components is equal to 
the weight W of the bar ; or, i^ + i^' = W. Furthermore, the moment 
of the force F about the point B (i.e. F x AB) will be found equal to 
the moment of the weight of the bar about the same point {i.e. W x 
CB); or, 

F X AB = W X CB. (3) 

The moment of the component F' about the point A will be found 
equal to the moment of W about the same point. Hence we may write 

F X AB = W X AC. (4) 

The laws of parallel forces may therefore be stated as 
follows : 




Fig. 



21. — Law of Parallel Forces 
Illustrated. 



LAWS OF MOTION — FORCE 41 

1. The resultant of two parallel forces acting in the same 
direction at different points on a body is equal to their sum, 
and has the saine direction as the components. 

2. The moment of one of the components about the point 
of application of the other is equal avid opposite to the moment 
of the supported weight about the same poiiit. 

Example. — Two men, A and B, carry a bucket weighing 100 lb. 
on a bar 10 ft. long. If the bucket is 4 ft. from A, how much force 
is exerted by each? 

Solution. — The moment of the force F exerted by A about the 
opposite end of the bar is 10 x F, and the moment of the weight about 
the same point is 100 x (10 — 4), or 600. Hence 10 x F= 600 ; whence 
i^ = 60 lb. Let F' be the force exerted by B. Then considering the 
moments about the other end of the bar, we have 10 x F' = 100 x 4 ; 
whence F' = 40 lb. Therefore A exerts 60 lb. and B 40 lb. 

47. The Couple. — When two equal parallel forces act 
upon a body along different lines and in opposite direc- 
tions, as shown in Fig. 22, they have no 
resultant; that is, no single force will have 
the same effect as the two components act- 
ing jointly. A combination of this kind is 
called a couple. The tendency of a couple is 
always to rotate the body on which it acts. 
This tendency is measured by the moment of 
the couple, which is the product of one of 
the forces multiplied by the perpendicular Fig. 22. — The 
distance AB between the two forces. This Couple, 
distance is the arm of the couple. The equilibrium of the 
body acted upon can be maintained only by the applica- 
tion of another couple of equal moment acting in the op- 
posite direction. 

A small magnet placed on a floating cork is rotated by 
the couple formed by the northward-acting force at one 
end and the equal southward-acting force at the other. 




42 A HIGH SCHOOL COURSE IN PHYSICS 

EXERCISES 

Note. — The student should first draw a diagram representing the 
conditions of the problem to be solved and then apply the general 
results deduced from the experiment in § 46. 

1. A uniform bar of wood weighing 12 kg. is 120 cm. long ; two 
hooks are placed on opposite sides of the center at distances of 40 cm. 
and 20 cm. respectively. What forces applied to the hooks will 
support the bar? Ans. 4 kg. and 8 kg. 

2. In order to support the bar in Exer. 1, what forces applied at 
points respectively 15 cm. and 15 cm. from the ends of the bar will 
be required? Ans. 9 kg. and 3 kg. 

3. A beam of uniform size is 60 ft. long and weighs 800 lb.; a 
man at one end supports 200 lb. Find the magnitude and point of 
application of the other required force. 

4. Two parallel forces of 30 g. and 70 g. are applied at the ends of 
a bar 1 m. long. Find what weight will be supported and its location 
on the bar, neglecting the weight of the bar itself. 

5. A boy and a man are carrying a weight of 150 lb. on a bar 10 
ft. in length. If the forces are applied at the ends of the bar, where 
must the load be placed in order that the boy may have to carry only 
50 lb. ? 

6. Draw a diagram showing a method for attaching three horses 
to a load so that they must pull equally. 

4. RESOLUTION OF FORCES 

48. Resolution of Forces. — In many cases it becomes 
desirable to find the effect of a force in some direction 
other than that in which it acts. For example, a car on a 

track running east and west, Fig. 
23, is acted upon by a force AB 
directed northeast. The given 
force has two effects : It produces 
(1) a tendency to move the car 
Fig. 23.— Force Resolved east, and (2) a pressure against 
into Two Components. ^j^^ j^j^j^g toward the north. It is 

plain that two forces, one directed east and the other 
north, might have the same effect as the single force 




LAWS OF MOTION — FORCE 



43 



directed northeast. In order to determine these two 
forces, the given force is represented hy the line AB^ from 
whose extremities, A and J5, lines are drawn completing 
the rectangle, as shown in the figure. AC is called the 
effective component, since it acts in the direction in which 
the car can move ; and AD is designated as the non- 
effective component, since it contributes nothing to the 
production of motion. 

A given force may he resolved into two components whose 
directions are given hy making the line of force the diagonal 
of a parallelogram whose sides are drawn from the point of 
application of the force in the directions required for the 
components. 

49. The Sailboat and the Aeroplane. — The principle of the 

resolution of forces explained above is readily applied to the opera- 
tion of a sailboat. Let the boat be headed north while the wind blows 
from the east. Now the pressure of the wind ^C on the sail SS\ 
(1), Fig. 24 can be resolved into AB, perpendicular to the sail, and a 





Fig. 24. 

second component AD, parallel to the sail, the latter of which is non- 
effective. Force AB is the effective pressure on the sail. If the 
vessel were round, it would move in the direction of AB. Now let 
AB be resolved as shown in (2), Fig. 24 into AE acting parallel to the 
keel and AE acting perpendicular to it. The former component 



44 



A HIGH SCHOOL COURSE IN PHYSICS 



Direction of Flight 



moves the vessel forward, while the component AE is rendered non 
effective by the deep keel of the boat. 

In the case of the aeroplane, which is a recent invention, huge 
planes or sails, ^^and CD, shown in the sectional view. Fig. 25, are 
attached firmly to a light frame, upon which is mounted a powerful 
gasolene motor (§ 271). The planes are slightly oblique, as shown. 
The power furnished by the motor turns a propeller whose office it is 
to drive the aeroplane rapidly forward. When the aeroplane moves 

forward to the left, it is as though a strong 
wind were blowing toward the right 
against the planes, as shown by the dotted 
lines. As in the case of the sailboat 
a pressure is produced at right angles 
to the planes AB and CD. Represent- 
ing this force by the line EF and re- 
solving it into two components, we find 
the lifting force EG, and the component 
EH, which tends to resist the forward mo- 
tion of the aeroplane. Smaller planes whose positions can be changed 
by the operator, are used in steering. 




Fig. 25. 



EXERCISES 

1. If the force represented by the line AB, Fig. 23, is 1000 lb., and 
the angle BA C, 60°, find the components AC and AD. 

2. Resolve a force of 2000 dynes into two components making 
angles of 30° and 60° with the given force. 

3. A weight of 50 kg. is suspended by two cords making angles of 
30° and 60° respectively with the vertical. Find the force exerted by 
each cord. Ans. 25 kg. and 43.3 kg. 

4. If the mass of the car in Exer. 1 is 20,000 lb., and the resistance 
offered by the rails may be neglected, what is the acceleration of the 
car? 

Suggestion. — Reduce the effective component to poundals and 
apply equation (2), § 36. Ans. 0.161 ft. per sec. per sec. 



5. CURVILINEAR MOTION 

50. Uniform Circular Motion. — It is a well-known fact 
that a ball attached to one end of a cord and whirled 
about the hand exerts a pulling force against the hand 



i 



LAWS OF MOTION — FORCE 45 

along the cord. This takes place because of the tendency 
of the ball to move in a straight line according to New- 
ton's First Law of Motion. In order, therefore, to confine 
the ball to a circular path, a continual force toward the 
center must be maintained. If this force is removed by 
the breaking of the cord, the ball will leave its circular 
path along a tangent. The force that continually deflects 
a moving body from a straight line^ compelling it to follow 
a curve^ is called centripetal force. If the motion of a 
body along the circumference of a circle is uniform, the 
centripetal force is constant. 

The name " centrifugal " force is often applied to the reaction of 
the moving body upon the fixed center. This reaction gives one the 
erroneous impression that the body would fly away from the center 
along a radius if the centripetal force should cease acting. If, how- 
ever, we watch the course taken by water or mud as it leaves a revolv- 
ing wheel, we readily observe that it moves along the tangent to the 
wheel at the poiut'where it is set free. 

51. Centripetal Acceleration. — Since a force produces 
a change in momentum in the direction of the force, 
according to Newton's Second Law (§ 35), a constant cen- 
tripetal force produces a constant change in the momen- 
tum of a body, which has uniform circular motion, toward 
the center. Since the mass is constant, the acceleration 
is constant and directed toward the center. If v is the 
velocity with which a body is moving in a circular path, 
and r the radius of the circle, the centripetal acceleration 
a is represented as follows :^ 

a = y- (5) 

1 Let a body m be moving around the circle whose center is o, with a 
uniform velocity v. Let it move from m to c in the very short time of t 
seconds. Then the distance mc will be equal to vt (Eq. 1, p. 14). If the 
time is small, the arc mc is practically equal to the chord mc. On com- 



46 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 26. — A Body m Having Circu- 
lar Motion Tends to Follow the 
Tangent to its Path. 



Thus, if the moving body is at the point m, Fig. 26, its tendency is 
to continue in the direction mh in accordance with the First Law of 

Motion. But, on account of the 
cord, it is compelled to keep the 
same distance from the center o and, 
consequently, is deflected from h to c. 
Again, at c, as at every point, the 
body tends to follow the tangent ce, 
but is compelled to take an inter- 
mediate path along the circumfer- 
ence to /. If the deflecting force 
is removed at the time the body 
reaches /, it continues to move in 
the direction of its motion at that 
point; that is, along the line fg, 
which is tangent to the circle at /. 
It is the component of the motion 
ma produced by the centripetal force acting along mo whose accelera- 
tion is represented in equation (5). 

52. Equation of Centripetal Force. — A force is equal to 
the product of the mass of the body upon which it acts 
and the acceleration that it produces (§ 36). Hence 
in circular motion the centripetal force is the product of 

the mass m and the centripetal acceleration — . Therefore 

r 

the value of the centripetal force may be expressed as 
follows : 

pleting the small rectangle mbca, we have, since ca is a perpendicular 
dropped upon the hypotenuse of the right triangle mch, 

mc^ — ma x mh. (1) 

Now the distance the mass m is drawn toward the center by the con- 
stant centripetal force in the time t is be and equals m«. Since the motion 
toward the center is unifonnly accelerated, 

^=;i^= iaf2 (Eq. 3, p. 18). (2) 

Therefore, by substituting the values mc = vt and Hia = ^af^ in (1), we 
have vH^ = ^at^ x 2 r, where r is the radius of the circle. From this 

equation v"^ = ar, whence a = —. 

r 



LAWS OF MOTION — FORCE 47 

2 

Centripetal force = ?^. (6) 



It should be observed that this equation gives the force 
in absolute units only ; i.e. in dynes., when C. G. S. units 
are substituted, and in poundals^ when F. P. S. units are 
employed. 

53. Illustrations of Circular Motion. — Many examples 
of circular motion present themselves in everyday life. 
The bicycle rider must carefully govern his speed as he 
turns a corner on a slippery pavement lest the force re- 
quired to change the direction of motion be too great and 
the wheels slip sidewise. In the modern cream separators 
the denser portions of the milk are forced to the outside 
of a rapidly revolving bowl, while the lighter cream re- 
mains near the center and is forced out along the axis. 
Honey is extracted by rapidly whirling the uncapped comb 
in a machine. Centrifugal driers are used in laundries 
and factories for removing water from clothing, wool, etc. 
In the " loop the loop " apparatus a car rides safely along 
a track within a large vertical circle, its own tendency to 
follow a tangent keeping it pressed firmly against the 
rails. 

The motion of the bodies of the solar system illustrates 
the action of centripetal force on the grandest scale. The 
earth, for example, having an initial motion, tends to move 
in a straight line. However, the attraction of the sun, 
like a tense cord, holds it in its orbit. If this force should 
cease, the earth would at once move away into space along 
a tangent to its orbit. On the other hand, if it were not 
for the earth's motion along the curve, it would be drawn 
with accelerated motion into the sun. 

The spheroidal shape of the earth is supposed to be due 
to the tendency of matter to withdraw from the axis of 



48 A HIGH SCHOOL COURSE IN PHYSICS 

rotation. This tendency causes bodies to weigh about jsq 
less at the equator than at the poles. 

When the centripetal force is not sufficient to keep the 
parts of a revolving body in the required circular paths, 
serious results often follow. This is the case of bursting 
fly wheels and emery wheels in mills and factories. 

EXERCISES 

1. Explain why water will not fall from a pail whirled at arm's 
length in a vertical circle. 

2. How is the overturning of a car prevented, as it rapidly turns a 
curve ? 

3. Does the rotation of the earth affect the weight of bodies in this 
latitude? 

4. Account for the fact that the moon moves in an orbit around 
the earth. 

5. What keeps the earth in rotation on its axis ? 

6. Show by equation (5) that increasing the rate of rotation of the 
earth seventeen fold would cause bodies at the equator to " lose " their 
entire weight. 

7. A body whose mass is 50 g. moves in a circle whose radius is 
40 cm. with a velocity of 20 cm. /sec. What is the required centripe- 
tal force ? 

8. A stone leaves a sling with a velocity of 50 ft. per second. If 
the mass of the stone is 2 oz. and the radius of the circle 4 ft., what 
was the pull exerted on the cords of the sling ? 

Ans. 78.125 poundals. 

SUMMARY 

1. The momentum of a body is measured by the prod- 
uct of its mass and velocity. It is represented by the 
expression mv (§ 33). 

2. The term force is the name given to the cause that 
produces acceleration, retardation, or a change in the 
direction of the motion of a body (§ 34). 

3. Force is .measured by the change in momentum 
produced per second. The C. G. S. unit of force is the 



LAWS OF MOTION — FORCE 49 

dyne. The dyne is that force which, acting uniformly 
for one second, imparts one C. G. S. unit of momentum 
(§ 35). 

4. The equation of force is/= — , or/= ma (§ 36). 

c 

5. The absolute units of force are the dyne Siud poundal ; 
the gravitational units are the gram-weight and pound-weight, 
etc. In everyday use the gravitational units are called 
simply the "gram " and ''pound " (§ 37). 

6. A given force produces its own effect, whether act- 
ing alone or conjointly with other forces (§ 38). 

7. To every action there is always an equal and oppo- 
site reaction; or, in other words, for every push or pull of 
one body upon a second body there is always an equal pull 
or push of the second body upon the first (§ 39). 

8. The characteristics of a force are its point of appli- \/ 
cation^ direction, and magnitude. Forces are represented by 
straight lines of suitable length and direction and may be 
compounded in the same manner as motions and velocities 

(§ 40). 

9. The resultant of two or more forces acting in the same 
direction along a straight line is equal to their sum; but 
when tAvo forces act in opposite directions in the same line, 
their resultant is equal to their difference and has the di- 
rection of the greater force (§ 42). 

10. The resultant of ttvo forces acting at an angle is rep- 
resented by the diagonal of a parallelogram constructed on 
the lines which represent the component forces. This law 
is universally known as the Principle of the Parallelogram 
of Forces (§ 43). 

11. The moment of a force about a point is the effective- 
ness of the force in producing a rotation. It is measured 
by the product of the magnitude of the force and the per- 

5 



50 A HIGH SCHOOL COURSE IN PHYSICS 

pendicular distance from the point to the line of direction 
of the force (§ 45). 

12. Any two parallel forces acting upward will support 
a weight equal to their sum, and the moment of one 
component about the point of application of the other is 
equal and opposite to the moment of the supported weight 
about the same point (§ 46). 

X 13. A system of two equal and opposite parallel forces 
acting along different lines is called a couple. The moment 
of a couple is the product of one of the forces multiplied by 
the distance between the two forces. A couple can be 
balanced only by another couple acting in the opposite 
direction and having an equal moment (§ 47). 

14. A force may be resolved into two components by 
making it the diagonal of a parallelogram whose sides are 
drawn in the directions required for the components (§ 48.) 

15. When a body has curvilinear motion, a force is re- 
quired to deflect the body continually from a straight line. 
This is called centripetal force. The equation of centripetal 

force is/ = — (§ 52). 



''HI 



CHAPTER IV 
WORK AND ENERGY 

1. DEFINITION AND UNITS OF WORK 

54. Work. — The use of the expression " to do work " 
is restricted in the study of mechanics to cases in which 
a force produces motion in the body upon which it acts. 
For example, attempting to lift a stone from the ground 
without succeeding in moving it is not doing work in the 
scientific sense ; but lifting the stone to a higher position 
implies that work is being done upon it. Similarly, bend- 
ing a bow is doing work, but holding it in a bent condi- 
tion is not. Lifting a weight involves the process of 
doing work, but simply supporting it does not. Work 
is done when the spring of a clock is wound, or a body is 
moved along upon a table. 

An important case in which work is done is that in 
which a freely moving body is given acceleration. We 
have already found (§ 34) that an increase in the velocity 
of a body requires the action of a force. Furthermore, 
the tendency of the force is to produce motion in the 
direction in which the force acts. Hence, work is done 
by exploding powder when it projects a bullet from a gun, 
or by gravity when a body is allowed to fall to the earth. 

55. Elements Involved in Work. — In each of the ex- 
amples given in the preceding section, it will be observed 
that two quantities are involved in the process of doing 
work. These are (1) the acting force and (2) the distance 
through which the force continues to act, sometimes called 
the displacement. Work is directly proportional to the force 

61 



52 A HIGH SCHOOL COURSE IN PHYSICS 

and the distance through tvhich the force acts, and is meas- 
ured hy their product. Thus 

Work = force x displacement. 

Or, if/ represents the force, d the distance through which 
the force acts. 

Work = fd. (1) 

56. Units of Work. — Since work is measured by the 
product of the force that acts upon a body and the dis- 
tance the body is moved in the direction of the force, a 
Ui^iit of tvork is done when a unit of force acts through a unit 
of distance. For every unit of force (§§ 36 and 37) there 
is a corresponding unit of work. The most important, 
however, is the C. G .S. unit which is called the erg. The 
erg is the work done ivhen a force of one dyne acts through a 
distance of one centimeter. The erg is also called the dyne- 
centimeter. 

The F. P. S. unit of work is the foot-poundal, which is the work done 
when a poundal of force acts through a distance of one foot. In the 
gravitational system two units are frequently used. The kilogram- 
meter is the work done when a force of one kilogram (§ 37) acts 
through a distance of one meter. The foot-pound is the work done 
when a force of one pound acts through a distance of one foot. 

The following table is given to show the use of equa- 
tion (l) in the computation of Avork in the different 
systems : 



Absolute System 

/ (in dynes) x d (in centimeters) = Work (in ergs) . 

/(in poundals) x d (in feet) = Work (in foot-poundals). 



Gravitational System 

/(in kilograms) x d (in meters) = Work (in kilogram-meters), 
/(in pounds) x d (in feet) = Work (in foot-pounds). 



WORK AND ENERGY 53 

Example. — A mass of 50 g. requires a force of 10 g. to overcome 
the friction and move the body at a uniform rate along a horizontal 
table. Find the work done when the mass is moved horizontally 
25 cm. Find also the work done when the mass is lifted 25 cm. 

Solution. — Since the horizontal force is 10 g. or 9800 dynes 
(§ 37). and the distance through which the force acts is 25 cm., the 
work performed is 10 x 25, or 250 gram-centimeters. Measured in 
ergs, the work is 9800 x 25, or 245,000 ergs. 

The work performed in lifting the mass is 50 x 25, or 1250 gram- 
centimeters. Measured in ergs, the work is 49,000 x 25, or 1,225,000, 
ergs. 

The numerical relation between the various units of 
work is shown in the following table : 

1 dyne-centimeter equals 1 erg. 

1 kilogram-meter (kg-m.) equals 98,000,000 ergs. 
1 foot-pound equals 13,550,000 ergs. 

1 foot-poiindal equals 421,390 ergs. 

2. ACTIVITY, OR RATE OF WORK 

57. Activity. — The value of any agent employed in do- 
ing work will depend upon the amount of work it is able 
to perform in a certain time. Some agents work slowly, 
others rapidly. For example, a man can lift a certain 
number of bricks to the top of a building in an hour ; a 
horse attached to a suitable hoisting mechanism can lift a 
greater number in the same time ; and an engine can lift 
the bricks as fast as several horses. Working agents are 
therefore said to differ in activity, or power. Activity is 
the rate of doing worh^ and is fou7id hy dividing the work 
performed hy the time consumed in the process. 

58. Units of Activity. — The unit of activity or power 
commonly used is the horse power (abbreviated H.P.). 
The horse power is the rate of doing work equal to 550 foot- 
pounds per second. The activity of an agent that is able 
to perform 550 foot-pounds of work per second is one horse 



54 A HIGH SCHOOL COURSE IN PHYSICS 

power. The unit of power in the C. G. S. system is the 
watt,^ which is equivalent to the work done at the rate of 
10^ ergs per second. One horse power equals 746 watts, 
or 746 X 10'' ergs per second. 

EXERCISES 

1. Calculate the work done by a force of 25 dynes acting through 
a distance of 120 cm. 

2. Express in ergs and gram-centimeters the work done in lifting 
a mass of 5 g. through a vertical height of 100 cm. 

3. A horse has to exert an average force of 200 lb. in moving a 
loaded cart a distance of a mile. Find the amount of work done. 

4. What amount of work is done when one cubic meter of water 
is elevated to a height of 10 m.? 

5. How much work is done per second by an engine that in one 
hour lifts 10,000 bricks each weighing 4 lb. to the top of a building 50 
ft. in height? Find the necessary horse power. 

6. A man shovels 3 T. of coal from a wagon box into a bin 6 ft. 
above the coal in the wagon. How much work is involved in the 
process? * 

7. What must be the power of an engine that hoists 50 T. of ore 
per hour from a mine 300 ft. deep? 

8. A pumping engine is capable of raising 300 cu. ft. of water 
every minute from a mine 132 ft. in depth. If a cubic foot of water 
weighs 62.5 lb., what must be the power of the engine? 

9. How long will it take a 3-H. P. engine to elevate 5000 bu. of 
wheat 50 ft.? (A bushel of wheat weighs 60 lb.) 

10. A train is moving with a velocity of 30 mi. per hour. If the 
resistance to the motion is 1500 lb., calculate the power utilized. 

11. The motors of an electric car can develop 200 H. P. With 
what velocity can the car run against a uniform resistance of 2200 lb. ? 

3. POTENTIAL AND KINETIC ENERGY 

59. Energy. — In each of the cases selected in § 54 to 
illustrate the process of doing work, some agent capable of 
doing work was assumed to be acting. The stone, for 

1 So called in honor of James Watt (1736-1819), the inventor of the 
steam engine. 



WORK AND ENERGY 55 

example, was supposed to be lifted by this agent, which 
may have been an engine, a person, a horse, or any other 
working medium. In order to be able to perform work, 
an agent must possess energy. The energy of a body is 
its capacity for doing work^ or its ability to do work. 

If we examine a body upon which work has been done, 
— any lifted mass, for instance, — we discover that the 
lifting process has invested the mass with the ability to do 
work upon some other body. The lifted body may be 
attached by a cord to the proper mechanism and allowed 
to fall back to its original position ; but during its fall it 
may turn wheels, lift another body, bend a bow, wind a 
spring, or do work in some other manner. Thus in doing 
work the falling body gives energy or working ability to that 
upon which the work is done. Similarly, a hammer by 
virtue of the velocity given it by the mechanic possesses 
the capacity for doing work. This is manifested by the 
fact that it drives the nail in opposition to the resistance 
offered by the wood. 

60. Potential Energy. — The lifted weight in the illus- 
tration used in the preceding section possesses energy be- 
cause of its elevated position. A bent bow has the ability 
to throw an arrow because of the fact that its form has 
been changed by some agent. The spring of a watch can 
keep the wheels moving against resistance on account of 
the fact that some one has done work upon it in winding 
it up. Energy possessed by a body because of its position or 
form is called potential energy. The potential energy of a 
body is measured by the work that was done upon it to 
bring it into the condition by virtue of which it possesses 
that energy. Hence a body whose mass is 100 pounds 
which has been lifted a distance of 5 feet has 500 foot- 
pounds of potential energy, i.e. it is able to do 500 foot- 
pounds of work because of its elevated position. 



56 A HIGH SCHOOL COURSE IN PHYSICS 

61. Kinetic Energy. — When a lifted weight is allowed 
to fall, it does work upon the object that it strikes. At 
the instant of striking it possesses energy because it is in 
motion. Moreover, any moving body is able to do work 
by virtue of its motion. The energy possessed hy a body 
because of its motion is called kinetic energy. The kinetic 
energy of a body is measured by the amount of work done 
upon it to put it in motion. 

62. Kinetic Energy Computed. — The kinetic energy of a 
moving body is measured by one half its mass multiplied by 
the square of its velocity. This may be shown in the fol- 
lowing manner : Let a body whose mass is m grams be 
acted upon by a force of / dynes which will give it an 
acceleration of a cm. /sec. 2. From (2), § 36, /= ma. 
Again, since the force produces uniformly accelerated 
motion in the given mass, at the end of t seconds, as 
shown by (3), § 25, the body will have been moved 
through a distance d = ^ af^. Now the velocity v 
acquired by the mass in t seconds as shown by (2), § 25, is 

V = at cm. /sec. ; whence f^ = — . Substituting this value 

a^ 

for t^ in the equation for distance, we obtain d = -- centi- 

2a 

meters. 

In order to compute the work done by the force /, 
we have only to multiply the force by the distance d 
through which it acts (§ 55^. Thus 



yZ yyfy 



Work =fd = ma X -— = --— ergs. 

2a 2 

Since the work done in producing the motion is the 
measure of the kinetic energy of the mass m (§ 59), 

Kinetic Energy = | mv^ ergs. (2) 



WORK AND ENERGY 57 

Let the mass, velocity, and acceleration be given in the units of 
the F. P. S. system. Then the product ma will give the force in 

poundals (§ 36), and the quantity — ■, the distance through which 

2 a 

the force acts, in feet. Therefore the product of force and distance 
will give the work in foot-poundals. Hence, in the F. P. S. system 
Kinetic Energy = \ mv^ foot-poundals. 
It should be remembered that the formula for kinetic 
energy deduced above gives the result in the absolute 
system only, i.e. in ergs or foot-poundals. These can be 
readily reduced to kilogram-meters and foot-pounds by 
the help of the numerical relations given in § bQ. 

Example. — Calculate the kinetic energy of a 10-gram bullet 
whose velocity is 40,000 cm./sec. 

Solution. — Using equation (2), we have for the kinetic energy 
of the bullet i x 10 x 40,000 x 40,000, or 8,000,000,000 ergs. By re- 
ferring to § 54, we observe that 1 kg-m. equals 98,000,000 ergs. 
Therefore, the reduction from ergs to kilogram-meters gives for the 
kinetic energy of the body 8163 kg-m. 

EXERCISES 

1. Calculate the potential energy given to a mass of 25 g. by 
lifting it through a vertical height of 10 m. Express the result in 
kilogram-meters. 

2. A ball moving with a velocity of 3500 cm./sec. has a mass of 
250 g. Find its kinetic energy in ergs. How much work must a 
boy do in order to stop it ? 

3. Compare the kinetic energy of the ball in Exer. 2 with that of 
a mass of 25,000 g. whose velocity is 350 cm./sec. 

4. What is the kinetic energy of a 5-gram bullet just as it is 
leaving the muzzle of a gun with a velocity of 30,000 cm./sec? 

5. To what height would the bullet in Exer. 4 have to be taken 
in order to have an equal amount of potential energy ? 

Suggestion. — First find the force required to lift the bullet in 
dynes; then apply equation (1), § 55. 

6. Compute the kinetic energy of a 5-pound mass moving with a 
velocity of 25 ft. per second. Express the result in foot-pounds. 

Suggestion. — First obtain the result in foot-poundals ; then re- 
duce to foot-pounds by the help of § 56» 



58 A HIGH SCHOOL COURSE IN PHYSICS 

7. A constant force of 200 dynes acts upon a mass of 5 g. Cal- 
culate (1) the acceleration, (2) the velocity produced in 3 seconds, 
and (3) the kinetic energy. What is the distance through which 
the force acts during the 3 seconds ? 

8. In order to move a load up a hill 250 ft. long, a horse exerts 
a constant pull of 125 lb. How much work is done? If the load 
weighs 900 lb., to what height would an equivalent amount of work 
lift it? 

9. A stone whose mass is 50 kg. is placed on the top of a chimney 
30 m. in height. Calculate the amount of work that must be per- 
formed in kilogram-meters and foot-pounds. 

10. Compute the amount of work done per minute by a pumping 
engine that forces 100,000 gal. of water into a reservoir 120 ft. high 
every 10 hr. Assume the density of water to be 62.5 lb. per cubic 
foot. 

11. If a rifle ball whose mass is 8 g. has a velocity of 35,000 
cm. /sec, how far will it penetrate a block of wood that offers a uni- 
form resistance of 100,000 g. 

Suggestion. — Let x be the depth of penetration in centimeters, 
and place the work done by the ball expressed in ergs equal to the 
kinetic energy. 

12. The elevation of a tank containing 25,000 gal. of water is 
75 ft. Find the potential energy of the water. 

4. TRANSITIONS OF ENERGY 

63. Transference and Transformation of Energy. — No 
processes in nature are of more common occurrence than 
transferences of energy from one body of matter to an- 
other and transformations of energy from one form into 
another. For example, if a body is allowed to fall freely, 
the potential energy that it possesses while elevated is grad- 
ually transformed into kinetic energy which resides in the 
body until its motion is checked. If, however, the body 
should fall upon a spring properly placed, the spring 
would be compressed and thus possess potential energy at 
the expense of the kinetic energy of the falling mass. 
Hence energy is transferred from the falling body to the 
spring. WTienever one body does work upon another^ energy 



WORK AND ENERGY 



59 




Fig. 27. — Transformation of the Po- 
tential Energy of the Raised Mass 
M into Kinetic Energy in the 
Wheel W. 



is transferred from the body that does the work to the one 
upon which the work is done. 

Let the elevated mass M, Fig. 27, be suspended by a cord wound 
around an axle A to which is at- 
tached a heavy wheel W. It is 
plain that the downward pull of 
M upon the cord will cause the 
wheel to turn. Thus, as M falls 
and loses potential energy, it 
does work upon the wheel in pro- 
ducing motion and thus impart- 
ing kinetic energy. 

When the cord is fully un- 
wound, the action will not cease ; 
but the kinetic energy of the 
wheel, by winding up the cord 
around the axle on the opposite 
side, will enable it to lift M. In 
this manner the wheel performs 
work upon M, losing its kinetic energy and imparting potential en- 
ergy to the mass lifted. If no energy were lost in overcoming fric- 
tion, the kinetic energy imparted to the wheel in the former case 
would be completely restored to the mass M in the latter. 

Changes in energy occar in a large number of common 
processes, such as winding a clock or a watch, shooting 
an arrow from a bow, running a sewing machine, turning 
a grindstone, running mills by water power, etc. 

64. Conservation of Energy. — Although energy is pass- 
ing continually through transformations and is being 
transferred from one body to another around us on every 
hand, no one has ever been able to prove that even the 
smallest portion can be created or destroyed. The infer- 
ence is, therefore, that the same quantity of energy is present 
in the universe to-day as existed ages ago ; i.e. that the quan- 
tity of energy present in the universe remains constant. 
This is known as the Law of the Conservation of Energy. 



60 A. HIGH SCHOOL COURSE IN PHYSICS 

The principal aim of Physics is to trace the various trans- 
formations and transferences of energy that accompany 
natural phenomena. At this point in the study many of 
these changes will seem obscure because all the forms in 
which energy may exist have as yet not been considered. 
For example, we may inquire what becomes of the kinetic 
energy of a spinning top as it slowly comes to rest. As 
we pursue the study further, we find that where there is 
motion in opposition to friction, as in this case, heat is 
produced. But heat is one of the forms that energy may 
take. Hence the kinetic energy of the top will appear 
somewhere in the form of heat. 

65. Matter and Energy. ^ — The intimate relation be- 
tween matter and energy is becoming more and more 
apparent. Matter is obviously a carrier, or vehicle, of en- 
ergy. We become acquainted with matter only through 
natural phenomena. In each phenomenon there is in- 
volved some change in energy, and it is in the transforma- 
tions and transferences of energy that our senses are 
affected. It is upon these processes that we base our 
entire knowledge of the material world. 

EXERCISES 

1. In driving a well a heavy w^eight is elevated by a horse and 
then allowed to fall upon the end of a vertical pipe, thus forcing it 
into the ground. Trace the energy changes taking place in the 
process. 

2. Trace the transferences and transformations of energy in the 
process of driving a nail; of planing a board; of shooting an 
arrow ; of throwing a stone ; of winding a clock ; of running a 
sewing machine ; of beating an egg. 

3. Account for the energy of the water above a dam in a river. 
What becomes of this energy ? 

4. In what form is a supply of energy taken on board an ocean 
steamer? In what form is energy supplied to a locomotive? to an 
automobile ? to a horse ? to a man ? 



WORK AND ENERGY 61 

SUMMARY 

1. The term work is used to express the process of pro- 
ducing motion. Work involves both force and motion in 
the direction of the force^ and is measured by the product 
of the force employed multiplied by the distance through 
which it acts. The equation of work is Work =/ x d 
(§ 55). 

2. The erg^ or dyne- centimeter^ is the C. G. S. unit of 
work and of energy and is the work done by a force of one 
dyne acting through a distance of one centimeter. The 
foot-poundal is the English absolute unit of work and en- 
ergy. The kilogram-meter Sind foot-pound are the gravita- 
tional units in common use (§ 56). 

3. The activity^ or power, of an agent is the rate at 
which it can do work. The activity of an agent is said to 
be one horse power when it can perform work at the rate 
of 550 foot-pounds (or 746 x 10'' ergs) per second (§§ 57 
and 58). 

4. The energy of a body is its capacity for doing work, 
or its ability to do work (§ 59). 

5. Potential energy is the energy possessed by a body 
because of its position or form (§ 60). 

6. Kinetic energy is the energy possessed by a body by 
virtue of its motion. The equation of kinetic energy is 
K.E. = 1 mv^ (§§ 61 and 62). 

7. When one body does work upon another, energy is 
transferred from the body that does the work to the one 
upon which the work is done (§ 63). 

8. Energy cannot be created or destroyed. The quan- 
tity present in the universe remains constant. This is 
known as the Law of the Conservation of Eyiergy (§ 64). 



CHAPTER V 
GRAVITATION 

1. LAWS OF GRAVITATION AND WEIGHT 

66. Universal Gravitation. — Ancient astronomical ob- 
servations revealed the fact that the planets move through 
space in curvilinear paths. Later and more refined obser- 
vations led to the discovery that the sun is a center about 
which they revolve in slightly elliptical orbits. Further- 
more, it is universally known that several of the planets 
have satellites which revolve about them, correspond- 
ing to the moon which moves in an orbit encircling the 
earth. Late in the seventeenth century Sir Isaac Newton 
originated the theory that is now known as the Law of 
Universal Gravitation, in order to account for the motion 
of heavenly bodies in nearly circular orbits instead of 
straight lines. 

67. Newton's Law of Universal Gravitation. — Tins law 
may be stated as follows : 

Every body in the universe attracts every other body with 
a force which is directly proportional to the product of the 
attracting masses and inversely proportional to the square of 
the distance between their centers of mass (§ 70). 

According to this law a book and a marble, or two 
bodies of any other kind of matter, attract each other. 
Between ordinary masses this force remains unnoticed by 
us in everyday life because it is so minute ; in fact, it 
would require the most refined test to detect it. But 
since the attraction is proportional to the .product of the 
masses^ i\\ the case of two heavy bodies — the moon and 

62 



GRAVITATION 63 

the earth, for example — the force is enormous. Even 
between the earth and a marble or a book the force is 
quite perceptible. When the earth is one of the acting 
masses, the attraction is called the force of gravity^ and 
when expressed in the proper units of measure, this attrac- 
tion is called the weight of the marble, book, etc. Weight, 
therefore, partakes of the nature of a force and is quite 
distinct from mass (§ 10). Hence, when we say in 
ordinary language, for instance, that the intensity of a 
certain force is 10 pounds, we mean that it is equal to 
that force witli which the earth attracts a mass of 10 
pounds. Again, since weight is a force, it may be ex- 
pressed in any of the units of force (§§ 36 and 37), i.e. in 
dynes, poundals, etc. 

68. Weight. — Since for a given locality the mass of 
the earth, as well as the distance from the center, is con- 
stant, the weight of a body is strictly proportional to its mass. 
Again, since the earth is not spherical but flattened 
slightly at the poles, the same mass at different places 
will not possess the same weight. On moving north or 
south from the equator the radius of the earth decreases 
slightly, which causes the mass to come somewhat nearer 
the earth's center and thus increases the value of the 
force of attraction. 

69. Law of Weight. — The law of universal gravita- 
tion applied to bodies outside the earth's surface is as 
follows : 

The weight of a body above the eartVs surface is inversely 
proportional to the square of its distance from the center of 
the earth. 

If the radius of the earth is assumed to be 4000 
miles, the weight of a one-pound mass 4000 miles 
above the surface, which is 8000 miles from the cen- 
ter, would be only one fourth of a pound. This result 



64 A HIGH SCHOOL COURSE IN PHYSICS 

is obtained by applying the law of weight as follows : 

X ; 1 pound : : 40002 . §0002 ; 

whence x ~ \ pound. 

Since the distance from the earth's center is less at 
Chicago than at the equator, a mass of 1 pound weighs 
about 5^0^ of a pound more at the former place than at 
the latter. For small differences of latitude, however, 
the difference in weight is so small that it is of little 
importance. In consequence of the earth's rotation the 
weight of bodies at the equator is diminished -^\-^ (§ 53) 
as the result of the centrifugal reaction against the force 
of gravity. In other latitudes this diminution is less. 

It is of interest to consider what the effect would be upon the 
weight of a given mass if it were to be taken to some point below 

the surface of the earth. Let it be im- 
agined that the circle in Fig. 28 repre- 
sents a cross section through the center 
of the earth, and that P is the location 
of the body to be weighed. All that 
part of the earth represented by the 
shaded portion of the circle above the 
plane AB will exert a resultant attrac- 
tion upward, while that represented by 
the unshaded part has a resultant acting 
Fig. 28. — A Mass atP is At- toward the center 0. Since these forces 

tracted Upward as well as oppose each other, the weight of the body 
Downward. .„,..., .. i .i - 

will dimmish as it approaches the center 

0. When the body reaches the center, the attractions due to the 

different portions of the earth will be equal in all directions, and the 

resultant of all will be zero. Therefore the body will weigh nothing 

at the center of the earth. 

EXERCISES 

1. If the mass of the earth were doubled without any change 
in its shape or size, how would a person's weight be affected? 

2. Which is a definite quantity, a gram of matter or a gram of 
force {i.e. a gram-weight) ? 




GRAVITATION 



65 



3. How much will the potential energy of a mass of 2000 lb. 
elevated 100 ft. at Chicago differ from that of an equal mass raised 
100 ft. at the equator ? 

4. A certain mass is weighed on a dynamometer (§ 11) at New 
York. Will the instrument indicate a greater or a less weight when 
the same mass is weighed at the equator ? 

5. If two masses are in equilibrium when placed in the pans of 
a beam balance at the equator, will they still be in equilibrium when 
tested in the same manner at San Francisco ? 

6. How far above the earth's surface would a body weigh one half 
as much as at the surface? Ans. 1656.8 mi. 

7. What would a 100-pound body weigh at a distance of 200 mi. 
above the earth's surface V 

8. An aeronaut ascends 5 mi. in a balloon. If his weight at the 
surface is 150 lb., what will it be at that height ? 




2. EQUILIBRIUM AND STABILITY 

70. Center of Gravity. — The weight of a body is the 
resultant of the weights of the individual particles of 
which it is composed. Since these 
innumerable forces are all directed 
toward the center of the earth, which 
is 4000 miles away, they form a sys- 
tem of essentially parallel forces 
whose resultant CA, Fig. 29, is equal 
to their sum (§ 46). The point of 
application O is called the center of 
gravity of the body. Since the posi- 
tion of this point depends upon the 
distribution of matter in the bod}^ it 
is also called the center of mass. In many problems it is 
convenient to consider the body as though all its mass 
were located at this point. If a flat piece of cardboard of 
any shape is balanced on the point of a pin, the center of 
gravity is located at the point of contact and midway 

between the two surfaces. 
6 



Fig. 29. — Weight is the 
Resultant of Innumer- 
able Small Forces. 



66 



A HIGH SCHOOL COURSE IN PHYSICS 




Let a flat piece of cardboard be pierced at any point, as A, Fig. 30, 
and hung loosely on a small nail or pin. The cardboard will turn 

until the center of gravity falls as low as pos- 
sible. In this condition a vertical line through 
A will pass through the center of gravity. 
This line is easily found by hanging a plumb 
line from the axis in front of the cardboard. 
If a second point of support, as B, be taken 
and a vertical line determined as before, the 
center of gravity C will lie at the point of in- 
tersection of the two lines. 
Fig. 30. — Locating the 

Center of Gravity. ^^ Equilibrium of Bodics. — A body 

is said to be in equilibrium when a vertical line through 
the center of gravity passes through a point of support. 
A common case of equi- 
librium is that of a chair 
or table. In such in- 
stances a vertical line 
through the center of 
gravity passes through 
the area of the base in- 
cluded within the lines 
joining the feet. Four 
in Fiof. 31. 




Fig. 31. — Stable, Unstable, and Neutral 
Equilibrium Illustrated. 



typical cases are represented 
Pyramid A hangs from its apex, B stands 
upon its base, and O rests with its apex at the point of 
support. Obviously A and B tend to remain indefinitely 
in the positions shown, but O will overturn with the 
slightest disturbance. If ^ or ^ should be tilted, the 
center of gravity would be lifted, necessitating the 
expenditure of energy upon the body. A and B are said 
to be in stable equilibrium. On the other hand, a disturb- 
ance of C lowers the center of gravity and thus lessens 
the potential energy of the body. When a body is in 
this condition, it is said to be in unstable equilibrium. 
A third condition is represented by a sphere of uniform 



GRAVITATION 67 

density resting upon a smooth horizontal plane. If the 
sphere be rolled along the plane, its center of gravity will 
be neither raised nor lowered. It is said to be in neutral 
equilibrium. A body arranged to turn upon an axis 
through its center of gravity is also in a condition of 
neutral equilibrium. 

72. Stability of Bodies. — When a body is in stable 
equilibrium, work must be performed upon it in order to 
cause it to overturn. This amount of work will depend 
upon the weight of the body and the distance through 
which its center of gravity is lifted, and is measured by 
their product (§ 55^. The amount of work required to 
overturn a body is a measure of its stability. 

Example. — Find the stability of a box 4 ft. square and 2 ft. high 
and weighing 500 lb. 

SoLUTiox. — Referring to Fig. 32, it is plain that the center of 
gravity C must be moved to the 

point A while the box is being over- ^^^^' \ 

turned. The height through which <" \ 

C is raised is BA, equal to OC — \ \ 

OB. Now OC is the hypotenuse \ , a \ 

of the right triangle OBC whose 

sides are 1 ft. and 2 ft. respectively. 

Hence OC equals \/5, or 2.24 ft. 

Therefore, AB is 1.24 ft., and the M MJMMM MMMMMB, 

work done 1.24 x 500, or 620 foot- Fig. 32.— Center of Gravity is Lifted 

pounds. Through the Height BA. 

It is clear that of two bodies having the same weight, 
the more stable one is that whose center of gravity has to 
be lifted through the larger vertical distance when we 
overturn it. This will depend on the size and shape 
of the base on which it rests and on the height of the 
center of gravity above the base. Figure 33 shows a brick 
in three possible positions. The center of gravity 
moves through an arc having the lower right-hand corner 



68 



A HIGH SCHOOL COURSE IN PHYSICS 



of the brick as its center when the body is overturned 
about this point. It will be seen at once that the greatest 



^-V— '" 



a I 
C / 



\ 



Vc 



(/) (2) {3) 

Fig. 33. — The Overturning of a Brick about Different Edges. 

stability is possessed by the brick when lying on its largest 
base ; first, because the base is largest, and second, because 
the center of gravity is in the lowest possible position. 
In each case c'a is the vertical distance through which the 
center of gravity would have to be lifted by the overturn- 
ing agent. 

Note the various methods employed to give the proper 
stability to objects in everyday use, as lamps, clocks, ink- 
stands, chairs, pitchers, vases, etc. 

EXERCISES 

1. How would you place a cone on a horizontal table in positions 
representing the three conditions of equilibrium ? 

2. Arrange two knives in a piece of wood as shown in Fig. 34 and 

support the point on the finger. Why is the 
system in stable equilibrium? Where is the 
center of gravity of the system ? 

3. Why is it difficult to walk on stilts? 

4. Explain the difficulty experienced in try- 
ing to balance an upright rod upon the end of 
the finger. 

5. Why does not the Leaning Tower of 
Pisa fall? See Fig. 36. 

6. Explain the difficulty experienced in trying to balance a meter 
stick on one end upon a level table. 




Fig. 34. — A System in 
Stable Equilibrium. 



GRAVITATION 



69 



7. The oil can B shown in Fig. 35 is loaded with lead at the bot- 
tom. Explain how this can will right itself while one of the common 
form A remains overturned. 

8. Which of two bodies hav- A 
ing equal weights possesses the 
greater stability, a pyramid or a 
rectangular box having the same 
base and height as the pyramid? Fig, 35. — Oil Cans. 

Suggestion. — The center of gravity of a pyramid is located at 
one third of the distance from the base to the apex. 

9. Calculate the stability in foot-pounds of a 4-pound brick placed 
in three different positions on a horizontal table. Assume the dimen- 
sions to be 2 X 4 X 8 in. See example in § 72. 




3. THE FALL OF UNSUPPORTED BODIES 

73. Falling Bodies. — Before the time of the Italian 
mathematician and physicist, Galileo Galilei,^ little 
was known concerning the way in which 
bodies fall when unsupported. It is a 
well-known fact that if we drop a coin 
and a piece of paper or a feather at the 
same instant, the coin will reach the floor 
first. Galileo rightly inferred that the 
difference was due to the resistance of- 
fered by the air. Nevertheless, in order 
to place on an experimental basis his 
conclusion that all falling bodies tend 
to have the same acceleration, he dropped 
bodies of different kinds from the top 
of the Leaning Tower of Pisa (Fig. 
36) in the presence of many learned men of the time. 
These experiments demonstrated that all bodies tend to fall 
from a given height hi practically equal times. Furthermore, 
it was readily shown that light materials, as paper, for ex- 
ample, fall in less time when compressed than when spread 
1 See portrait facing p. 70. 




Fig. 36. — Leaning 
Tower of Pisa, 
Italy. 



70 



A HIGH SCHOOL COURSE IN PHYSICS 




out. Later, after the invention of the air pump, Galileo's 
inference was verified by allowing a light and a heavy 
body to fall in a vacuum. For this purpose 
the " guinea and feather " tube (Fig. 37) is 
commonly used. On inverting the tube 
after the air has been exhausted, we find 
that the feather falls as rapidly as the coin. 
But when the air is again admitted, the 
feather flutters slowly along far behind the 
rapidly falling coin. 

74. Uniform Acceleration of Falling Bodies. 
— Since the attraction existing between a 
body and the earth is constant, it follows that 
a body falling freely, i.e. without encoun- 
tering resistance, will have uniformly accel- 
^Y erated motion (§§24 and 36). Again, since 
Fig. 37. — Bodies all bodies fall the same distance in a given 
Fall Alike in time wheu unimpeded, the acceleration will 
be the same for all bodies. This acceleration 
is called the acceleration due to gravity, and is designated 
by the letter g. 

75. The Acceleration Due to Gravity. — The acceleration 
of freely falling bodies varies according to the laws of 
weight given in § 69. Since the distance to the center of 
the earth decreases as one travels from the equator toward 
the poles, the acceleration due to gravity becomes greater. 
In latitude 38° N. g is 980 cm./sec.2, and in latitude 50° N. 
981 cm./sec.2. The value of g also decreases slightly with 
the elevation above sea-level. (Why ?) In the latitude 
of New York, 40.73° N., a freely falling body gains in ve- 
locity at the rate of about 980 centimeters, or 32.16 feet, 
per second during each second of its motion. 

When bodies are thrown upward, the acceleration is negative ; i.e. 
the velocity decreases at the rate of 980 centimeters per second 




GALILEO GALILEI (1564-1643) 



The first successful experimental investigations relating to falling 
bodies and the pendulum must be attributed to Galileo. For nearly 
twenty centuries the science of Mechanics had remained undeveloped. 
Aristotle had announced that the rate at which a body falls depends 
upon its weight, but Galileo was the first to disprove it by experi- 
ment. This he did by dropping light and heavy bodies from the 
leaning tower of Pisa, Italy, his native town. A one-pound ball and 
a one-hundred-pound shot, which were allowed to fall at the same 
time, were observed by a multitude of witnesses to strike the ground 
together. Hence the rate of fall was shown to be independent of 
mass. 

At another time, while observing the swinging of a huge lamp 
in the cathedral, Galileo was astonished to find that the oscillations 
were made in equal periods of time no matter w^hat the amplitude. 
He proceeded to test the correctness of this principle by timing the 
vibrations with his own pulse. Later in life he applied the pendu- 
lum in the construction of an astronomical clock. 

Galileo was the first to construct a thermometer and the first to 
apply the telescope, which he greatly improved, to astronomical 
observations. He discovered that the Milky Way consists of innu- 
merable stars; he first observed the satellites of Jupiter, the rings of 
Saturn, and the moving spots on the sun. 

Galileo was made professor of mathematics in the University of 
Pisa in 1589 and filled a similar position at Padua from 1593 until 
1610. He died in the year 1643, the year of Newton's birth. 



GRAVITATION 



71 



during each second of its upward motion. Hence, a body thrown 
upward with a velocity of 2940 centimeters per second will continue 
to rise 3 seconds, when it will stop and return to earth in the next 3 
seconds. However, on account of the hindrance of the air, bodies 
moving with great velocity deviate considerably from the laws govern- 
ing unimpeded bodies. 

76. Laws of Freely Falling Bodies. — Since the motion 
of an unimpeded body while falling is uniformly acceler- 
ated, the equations of § 25 may be applied by simply sub- 
stituting for a the acceleration due to gravity g. These 
equations may be written as 
follows : 

V = gt, (1) 



and 



gt2. 



(2) 



dforl sec =1 x%g 
AtB, 



'2- 
v=g 



From equations (i) and (2) 
other useful formulae may be 
deduced. From (l) we find that 

t^ = —' Substituting this value 
for ^ in equation (2), we obtain 

Tr2 



dfor2 sec^ItX ^/ig 



Ate, ^^2g -- 



d = 



2g 



y ) dfor 3 sec=9x%g 



Solving equation (3) for v^ 
we have 

(4) 



AtD, v=3g 



dforh sec- 



-16 xV^ 



'2 9 



>3x/29 



>5x 



V2 



1 x/2 9 



V2i 



'Ag 



t 7 x/2Q 



V2 



V = V2 gd. 
77. Distances and Velocities 
Represented. — The distances 
passed over and the velocities 
acquired by a freely falling body 
are represented graphically in 
Fig. 38. A vertical line is 
drawn on which a convenient distance AB is measured off 
to represent 1 x J^ (about 16 feet), the distance the 
body falls during the first second. The distance J. (7 is 



Fig. 38. — Motion of a Falling 
Body Represented. 



72 



A HIGH SCHOOL COURSE IN PHYSICS 



made four times the distance AB ; AD, nine times AB ; 
AU, sixteen times AB, etc., to represent the distances 
fallen in one, two, three, and four seconds respectively. 
The heavy arrows are drawn to represent the velocity 
at the end of each second. The length of the first is 
g units (representing about 32 feet per second), the 
second 2^, the third 3^, etc. 

78. Bodies Thrown Horizontally. — It is a well-known 
fact that a body projected in any direction except up or 
down follows a curved path. An interesting case of this 

kind is the projection of a body 
horizontally from some elevated 
position, as A, Fig. 39. The 
motion of the body will be the 
resultant of two component mo- 
tions, the one vertically down- 
ward due to gravity, and the 
other in a horizontal direction 
due to the projectile force. 
Since there is no horizontal 
force acting on the body after it 
leaves the point A, the horizon- 
tal component of the motion 
will be uniform, and the body 
will move (horizontally) over 
equal distances in equal intervals 
of time. Let AB', B' C, CI)', etc., represent these hori- 
zontal distances for successive seconds. The distances AB, 
BO, CD, etc., are drawn in the manner described in § 77. 
Under the combined action of its initial horizontal velocity 
and the force of gravity the body will pass through the 
point Pj at the end of the first second, P^ at the end of the 
second, Pg at the end of the third, etc. Thus the body 
follows the curved path AP^P^P^P^. 




Fig. 39. — Motion of a Body Pro- 
jected Horizontally from the 
Point A. 



GRAVITATION 73 

The path of a stone thrown over a tree, for example, is a case in 
which the initial motion of the body is not horizontal. The motion 
may be divided into two parts : first, the rise of the stone to the highest 
point reached ; and, second, the fall of the body back to the earth. 
The latter half of the motion is precisely the case described above. 
The time during which the stone is rising is practically equal to that 
of its fall. The time required for the return of the stone to the earth 
is the same as that of a body falling vertically. Likewise the time 
occupied by the stone in rising is equal to that required by a body 
thrown vertically upward to attain the same height. The shape of 
the path taken by the stone depends on its initial speed and the direc- 
tion in which it is thrown. The study of the motion of projected 
bodies, as bullets, cannon balls, shells, etc., forms an important part of 
military and naval instruction. 



EXERCISES 

1. A body falls freely from a certain height and reaches the 
ground in 5 seconds. What" velocity is acquired? From what height 
must it fall ? 

2. How long does it take a body to fall 100 ft. ? 200 ft. ? 

3. A mass of 50 g. falls for 3 seconds from a state of rest. Calcu- 
late its kinetic energy. (See § 62.) 

4. A mass of 50 lb. falls from an elevation of 20 ft. Calculate its 
kinetic energy in foot-pounds at the time it reaches the ground. Com- 
pare the kinetic energy with the potential energy of the body before 
falling. 

5. How far must a body fall in order to acquire a velocity of 
500 ft. per second ? 

6. A book falls from a table 3 ft. in height. Find the velocity of 
the book when it reaches the floor. 

7. A stone is dropped from a train whose velocity is 30 mi. per 
hour. Show by a diagram the path traced by the stone. (See § 78.) 

8. Find the kinetic energy of a lO-gram mass after it has fallen 
from rest a distance of 1960 cm., assuming g to be 980 cm./sec.2. 

9. The velocity of a body falling freely from rest was 200 ft. per 
second. From what height did it fall? 

10. Compare the velocity of a body after falling 64.32 ft. with 
that of a train running 30 mi. per hour. 

11. A bullet is fired vertically upward from a gan with a velocity 



74 



A HIGH SCHOOL COURSE IN PHYSICS 



of 25,000 cm./sec. Disregarding the resistance of the air, how many 
seconds will the bullet continue to rise? How high will it rise? 

12. If the bullet in Exer. 11 encountered no resistance due to the 
air, how many seconds would pass before it returned to earth ? 

13. The weight of a pile-driver is lifted 10 ft. and allowed to fall. 
With how much greater velocity will it strike if lifted 20 ft. ? With 
how much greater energy ? 

14. A stone thrown over a tree reaches the earth in 3 seconds. 
What is the height of the tree? 

15. A boy fires a rifle ball vertically upwards and hears it fall upon 
the ground in 20 seconds. How high does it rise? What was its 
initial velocity? 

4. THE PENDULUM 

79. The Simple Pendulum. — A heavy particle sus- 
pended from a fixed point hy a weightless thread of constant 

length is an ideal simple pendu- 
lum. If, however, we suspend 
a small metal ball A^ Fig. 40, 
by a thin flexible thread or wire 
from a fixed point 0, it fulfills 
the ideal conditions almost per- 
fectly. The distance OA from 
the point of suspension to the 
center of the ball is the leiigth 
of the pendulum. The arc AO^ 
or the angle A 00^ which repre- 
c'('t^ /^^c sents the displacement of the 

9 ball from the position of equi- 

librium, is the amplitude of vi- 
bration. A vibration is one to- 
and-fro swing, sometimes called a complete or double vibra- 
tion. A single vibration is the motion of the ball from Q 
to C ., or one half of a complete vibration. The period of 
a siyigle vibration is the time consumed by the pendulum 
in moving from (7 to (7', 



Fig 40. — The Simple Pendulum. 



GRAVITATION 



75 



^ 



75^-- ,j "'5' 



P' 



Fig. 41. 




Gravity Causes a Pendulum 
to Vibrate. 



80. Pendular Motion Due to Gravity. — When a pen- 
dulum is in the position of rest, the weight of the ball rep- 
resented by the line A "FT, Fig. 41, is balanced by the equal 
and oppositely directed ten- 
sion AT in the cord OA. 
Now let the ball be drawn 
aside to the position C and 
released. The weight of the 
ball, which always acts ver- 
tically downward and is 
represented by the line CIP, 
can be resolved into two 
components. One of these 
components is represented 
by the line CE and serves 
solely to produce tension in 
the cord CO. It is plain that this component has no effect 
on the motion of the ball. The other component CB acts 
upon the ball along the tangent to the arc of vibration at 
the point C. The effect of this component is to give the 
ball accelerated motion along the arc. After the ball passes 
the point A^ it is clear that the component along the tangent 
tends to retard the motion and finally succeeds in stopping 
the ball at the point C, after which it returns the ball 
again to A. 

If the line CD is drawn perpendicular to OA, the triangles 
CDO and CBP are similar. Why? We may therefore write the 
proportion 

CB : CP : : CD : CO (5) 

This proportion may also be written as follows : 

Force CB = displacement CD x weight of ball CP . .^^ 

length CO ^ 

Since the weight of the ball CP and the length of the pendulum 
CO remain constant during a vibration, equation (6) shows that the 
effective force CB is proportional to the displacement CD. Hence 



7G 



A HIGH SCHOOL COURSE IN PHYSICS 



the acceleration of the ball is not the same at all points in the arc 
CA, but varies directly as the displacement. 

81. Transformations of Energy in the Pendulum. — A 

pendulum is iirst set in motion by displacing the ball, or 
pendulum bob. This process requires the performance of 
work which elevates the ball through the height AD^ Fig. 
41 (measured vertically), and stores potential energy in it. 
From equation (i), § 55, it is clear that the amount of 
potential energy given the ball is measured by the product 
of its weight and the height AD. As the ball moves toward 
A^ velocity is acquired, and the potential energy is gradu- 
ally changed into kinetic. At A the energy is all kinetic. 
After passing the point A the ball rises to C, while the 
kinetic energy is transformed back into potential. At C 
the energy is all potential again. Thus recurrent trans- 
formations of energy take place, which would occur with- 
out loss if it were not for the resistance offered by the 
air as well as by friction at the point of 
suspension, 

82. Laws of the Simple Pendulum. — 
The first three laws of the simple pendu- 
lum may be deduced from the results 
obtained from the following experiments : 

Suspend four balls as shown in Fig. 42. Let 
A, B, and C be of metal, and D of wood or wax. 
Make the lengths ot A, B, and C as 1:4:9; e.g. 
20, 80, and 180 cm. Also let D be made precisely 
of the same length as C. Now if C and D be set 
swinging through the same amplitude, it will be 
readily observed that the period of vibration of D 
is the same as that of C 

Again, let C and D be set in vibration through 
different amplitudes. If neither amplitude is large, 
it will be seen that the period of one is still the 
same as that of the other. 

Finally, let A, B, and C be put in motion successively and the 




Fig. 42. — Pendu- 
lums of Differ- 
ent Lengths and 
Masses. 



GRAVITATION 77 

single vibrations of each counted for one minute. If the period of 
vibration of each pendulum be computed from the number of vibra- 
tions per minute, it wiJl be found that the three periods are as 1 : 2 : 3, 
i.e. as VT : Vi : V9. 

The laws governing the vibration of simple pendulums, 
therefore, may be stated thus : 

(1) The period of vibration is independent of the material^ 
or mass^ of the ball. 

(2) When the amplitude of vibration is small., the period 
of vibration is independent of the amplitude ; i.e. the vibra- 
tions are made in equal times. This is called the Law 
of Isochronism (pronounced i soJc'ro nisnfi). If the ampli- 
tude exceeds 5° or 6°, the period of vibration will gradually 
diminish as the arc becomes smaller. 

(3) The period of vibration is directly proportional to 
the square root of the length of the p)e7idulum. This is 
called the Law of Length. This law may be represented 
thus : t^: t^'. : VZj : VZgi where t^ and l^ refer to the period 
and length of one pendulum, and ^2 '^^^^ ?2' ^^ ^^^^ period 
and length of the other. If, therefore, any three of the 
terms are given, the fourth may be computed. 

Since a pendulum is dependent upon the force of gravity, 
as shown in § 80, its period of vibration is found to depend 
upon the value of g. Hence : 

(4) The period of vibration is inversely proportional to 
the square root of the acceleration due to gravity. 

83. The Pendulum Equation. — The relation between 
the period and length of a simple pendulum and the 
acceleration due to gravity is given by the equation 

t='n'Vl (7) 

g 

where t is the period of a single vibration, I the length of 
the pendulum measured in centimeters (or feet), and g 



78 



A HIGH SCHOOL COURSE IN PHYSICS 



the acceleration due to gravity measured in centimeters 
per second per second (or feet per second per second). 
The value of ir is 3.1416, the ratio of the circumference 
of a circle to the diameter. 

This equation is of great assistance (1) in calculating 
the period of any simple pendulum of known length, (2) 
in determining the length of a pendulum that vibrates in 
any given period of time, and (3) in finding the value of 
g at any place where its magnitude is unknown. (See § 87.) 
84. The Seconds Pendulum. — A pendulum of which the 
period of a single vibration is one second is a seconds 
pendulum. As shown by equation (7), the length of a 
simple pendulum Z, when the period t is 
one second, will depend on the acceleration 
due to gravity at the place chosen. At all 
places where the value of g is 980 cm. /sec. 2, 
the length Z of a simple pendulum that beats 
seconds is 99.3 cm.; where ^ is 981 cm. /sec. 2, 
I is 99.4 cm. 

85. The Compound Pendulum. — When 
the conditions defining the ideal simple pen- 
dulum (§ 79) are not sufficiently fulfilled, 
the body is called a compound or physical 
pendulum. For experimental purposes a 
meter bar may be suspended on a smooth 
wire nail which pierces it at right angles 
close to one end. The bar may be hung to 
swing freely between the prongs of a large 
tuning fork, as shown in Fig. 43. Let a 
simple pendulum OA be placed by the side 
Simple Pendu- q£ ^^iq suspended bar so that their points of 



Fig. 43. — A Com- 
pound Pendu- 
lum and its 
Equivalent 



lum 



suspension lie in the same horizontal plane. 
Set both pendulums in vibration and adjust the length of 
the simple one until they vibrate in the same period of 



GRAVITATION 79 

time. It will be observed at once that the simple pendu- 
lum must be made several inches shorter than the other. 
Only those points in the compound pendulum very near 
the point C swing in their natural period ; particles below 
C tend to swing slower, and those above, faster, than the 
simple pendulum. (7 is called the center of oscillation of 
the compound pendulum. In this case the point c is lo- 
cated two thirds of the length of the bar from the point 
of suspension. The simple pendulum is thus seen to be a 
special case of the compound one in which the entire mass 
is concentrated near the center of oscillation. 

The compound pendulum in Fig. 43 may be set swinging 
by being struck a sharp blow at the point C, and the axis 
will not be disturbed. For this reason Q is called the center 
of percussion. Thus, for example, when a ball is batted, 
the bat should be so handled that the ball will strike its 
center of percussion. This will prevent the jarring of the 
hands and the breaking of the bat. 

86. The Compound Pendulum Reversible. — If the meter 
bar shown in Fig. 43 be suspended by piercing it at the 
center of oscillation O and swinging it about this point, 
the period of vibration will be the same as before. In 
other words, the point of suspension B and the center 
of oscillation Q are interchangeable. This property 
of a compound pendulum, which was discovered by the 
famous Dutch physicist Huyghens (1629-1695), is 
known as its reversibility. 

87. Utility of the Pendulum. — The value of the pendu- 
lum in the measurement of time is due to the isochronism 
of its vibrations. Although Galileo was the first to ob- 
serve this property of the pendulum and the first to make 
a drawing of a pendulum clock, Huyghens was the first to 
use a pendulum in controlling the motion of the wheels of 
a timepiece. This he accomplished in 1656. 



80 



A HIGH SCHOOL COURSE IN PHYSICS 



The motion of a clock is maintained by lifted weights or by the 
elasticity of springs. The office of the wheelwork is to move the 
hands over the dial and to keep the pendulum from 
being brought to rest by friction. The latter is 
effected by means of the escapement shown in 
Fig. 44. The wheel R is turned by mechanism 
not shown in the figure. When the pendulum 
swings to the right, motion is communicated to the 
curved piece MN through the parts A, B, and O ; 
and AI is lifted. The wheel is thus released ; but, 
on turning, strikes at N. While the pendulum 
moves to the left, the slight pressure of the cog 
against N causes A to deliver a minute force to the 
pendulum. As iV rises, the wheel is again re- 
leased, but is again detained at M. Thus one cog 
is allowed to pass for each complete vibration of the 
pendulum. Every " tick " of the clock is caused 
by the wheel R being stopped either at M or N. 
If the wheel is allowed to turn too fast, the clock 
gains time. This defect is corrected by lowering 
the bob. If the clock loses time, the bob is raised. 

The pendulum offers the most precise 
method for measuring the acceleration of 
gravity. By carefully determining experi- 
ment and Pen- mentally the period t and the length Z of a 
duium of a Clock, pendulum, the value of g can be easily cal- 
culated by the help of equation (7), § 83. 

EXERCISES 

1. By the help of equation (7), § 83, find the period of a pendulum 
80 cm. long, when g equals 980 cm./sec.^. 

2. Calculate the length of a simple pendulum that beats half 
seconds (i.e. t equals | sec.) at a place where the acceleration is 981 
cm./sec.2. 

3. The pendulum of a clock has a period of a quarter second. 
Find its length if g is 980 cm./sec.^. 

4. How long is a simple pendulum that makes 65 single vibrations 
per minute? 

Suggestion. — Fu"st compute the value of t. 




GRAVITATION 81 

5. What is the value of g where a simple pendulum 99.2 cm. long 
makes 60 single vibrations per minute ? 

6. An Arctic explorer finds that the length of the seconds pendu- 
lum at a certain place is 99.6 cm. What is the value of g at this place ? 

7. A simple pendulum is to make 45 single vibrations per minute. 
If g is 980 cm./sec.2, what must be its length? 

8. It is found at a certain place that a simple pendulum 90 cm. 
long makes 64 single vibrations per minute. Find the value of g at 
this place. 

9. A pendulum whose bob weighs 100 g. is drawn aside until the 
distance AD, Fig. 41, is 4 cm. How much energy is stored in the 
bob ? How much work was done upon it ? 

SUMMARY 

1. Newton's Laiv of Universal Gravitation states that 
every body in the universe attracts every other body with 
a force that is directly proportional to the product of the 
attracting masses and inversely proportional to the square 
of the distance between their centers of mass (§ 67). 

2. The attraction of the earth for other bodies is called 
the force of gravity/. The weight of a body is the measure 
of this force (§ 67). 

3. The weight of a body is proportional to its mass 
(§ 68). 

4. The weight of a body above the earth's surface is in- 
versely proportional to the square of its distance from the 
center of the earth. On account of the spheroidal form of 
the earth, a body at the equator weighs slightly less than 
a body of the same mass at some other point on the earth's 
surface (§ 69). 

5. The weight of a body is the resultant of the weights 
of the individual particles that compose it. The point of 
application of this resultant is the center of gravity of the 
body (§ 70). 

6. A body is in equilibrium when a vertical line drawn 
through its center of gravity passes through a point of 



82 A HIGH SCHOOL COURSE IN PHYSICS 

support, or within the area included between the extreme 
points of support. The three kinds of equilibrium are 
stable^ unstable^ and neutral (§ 71}. 

7. The stability of a body is measured by the work that 
must be performed in order to overturn it (§ 72). 

8. All freely falling bodies descend from the same 
height in equal times. Such bodies have uniformly ac- 
celerated motion. The acceleration due to gravity is about 
980 cm./sec.2, or 32.16 ft./sec.2 (§§ 73-75). 

9. The equations of freely falling bodies are (1) v = gt^ 

(2) d = 1^^^ (3) (^ = — , and (4) v = V2^ (§ 76). 

1g 

10. When bodies are thrown, the horizontal component 
of the motion is uniform, while the vertical component is 
uniformly accelerated (§ 78). 

11. The swinging of a pendulum is due to the force of 
gravity (§ 80). 

12. The period of vibration of a pendulum is independ- 
ent of the mass of the bob and the amplitude of vibra- 
tion when the arc is small, and is directly proportional to 
the square root of the length and inversely proportional 
to the square root of the acceleration due to gravity. The 

equation of the pendulum is ^ = tta/- (§ 82). 

13. A compound pendulum may be conceived as being 
made up of simple pendulums of different lengths. The 
period of vibration depends upon the position of the center 
of oscillation. The center of oscillation and point of sus- 
pension are interchangeable. The center of percussion 
and the center of oscillation coincide (§§ 85 and 86). 



CHAPTER VI 
MACHINES 

1. GENERAL LAW AND PURPOSE OF MACHINES 

88. Simple Machines. — The transference of energy from 
a body capable of doing work to another upon which the 
work is to be done is often accomplished more advanta- 
geously by the use of a simple machine than in any other 
way. Indeed, it is often impossible for an agent to do 
the required work without the aid of a machine. For 
example, a man wishes to load a barrel of lime into a 
wagon, but finds that he is unable to lift it ; with the 
aid of an inclined plane of suitable length, however, the 
barrel is easily rolled into the wagon. In other cases use 
is made of the pulley^ lever, wheel and axle, screw, and wedge, 
which with the inclined plane form the six simple machines. 

89. The Principle of Work. — The 
general law of machines is illustrated by 
the following experiments : 

1. Let a cord be passed over a pulley, as shown 
in Fig. 45. Let the pull of a dynamometer be 
used to counteract the weight of the body W. 
It is obvious in this case that the amount of force 
F registered on the dynamometer must be equal 
tp the weight W. The experiment will show Fig. 45.— The Effort 
that this is the case. Furthermore, if force F i^ Equals the 
moves downward 1 foot, W will be elevated MovSThrLgh an 
through an equal distance. Equal Distance. 

If, now, we designate the distance through which the 
acting force (or effort) F moves by the letter d and the 

83 




84 A HIGH SCHOOL COURSE IN PHYSICS 

distance the weight Wis lifted by d\ the work put into the 
machine by the acting agent is -F x c?, and the work done 
by the machine is W x d' . It is plain that the experiment 
shows that Fxd=Wxd'. 

2. Let the pulley be now attached to the weight W, Fig. 46, and 
let one end of the cord be fastened to some stationary object at ^. If 
W is made 1000 grams, for example, it will be 
found that the upward effort registered by the 
dynamometer will be 500 grams. For any value 
of W, F will be one half as great. However, when 
F moves a distance of 1 meter, for example, W is 
elevated only one half a meter. 




In this experiment W=2F^ and d' = ^d* 
Therefore, we may write as before 

F X d = W X d^ (1) 

The relation shown by this equation is 
one of the most important laws of me- 
FiG. 46. — TheEf- clianics and is known as the Principle of 
One Half ofThe Woik. The principle may be stated as 

Weight W, but folloWS: 

Moves Twice as 

Far. 2^}ie work done hy an ageMt upon a ma- 

chine is equal to the work accomplished hy the machine ; or^ 
the effort F multiplied hy the distance through which it acts 
equals the resistance W that is overcome hy the machine mul- 
tiplied hy the distance it is moved. 

This law may be applied to any machine, no matter how 
simple or complicated it may be, provided the friction of 
the moving parts can be disregarded. 

Example. — An agent capable of doing work exerts a force of 
50 lb. upon a machine. If a weight of 250 lb. is lifted 8 ft. by the 
machine, through what distance must the applied force act ? 

Solution. — The work done by the machine is 250 x 8, or 2000 
foot-pounds. Hence, the force applied by the agent must act through 
the distance 2000 -- 50, or 40 ft. 



MACHINES 85 

It is clear from this example that the gain in force is 
accomplished at the expense of distance, since the effort 
must move five times as far as the resistance. Its speed 
also is five times as great. On the other hand, a machine 
may be made to increase the distance as well as the speed 
at the expense of force. Such is the case in many practi- 
cal applications of the simple machines. 

90. Mechanical Advantage of a Machine. — It is fre- 
quently desirable to know the multiplication of force that 
is brought about by the use of a machine ; or, in other 
words, the ratio of the resistance W to the effort F. This 
ratio is called the mechanical advantage of the machine. 

From equation (l) we may write 

W:F::d:d'. (2) 

Hence, tJie mechanical advantage of a machine is the ratio 
of the distance through which the effort moves to the distance 
through which the resistance is moved hy the macJmie. 

For example, the mechanical advantage of the pulley as 
used in the first experiment of the preceding section is 1, 
in the second experiment 2, and in the example given 
on page 84 it is 5. 

2, THE PRINCIPLE OF THE PULLEY 

91. The Pulley. — The pulley consists of a grooved 
wheel, called a sheave, turning easily in a block that 
admits of being readily attached to objects. When the 
block containing the sheave is attached to some stationary 
object, the pulley is said to be fixed; when the block 
moves with the resistance, the pulley is movable. A fixed 
pulley is shown in (1), Fig. 47, and a single movable 
pulley in (2). 

Let experiments be made with pulleys arranged as shown in Figs. 
47 and 48. In each case ascertain by means of a dynamometer or 



86 



A HIGH SCHOOL COURSE IN PHYSICS 



weights the force requn-ed 
at F to balance a weight 
applied at W. The effect of 
friction may largely be 
avoided by taking the mean 
of the forces applied at F, 
first when W is slowly raised 
and then when it is slowly 
lowered. 

When a single fixed 
pulley is used, W= F. 
When a single movable 




(0 





(.2) 




Fig. 47.— (1) A Fixed Pulley. 
(2) A Single Movable Pul- 
ley. (3) A System of One 
Fixed and One Movable 
Pulley. 

pulley is used, the re- 
sistance is applied to 
the movable block, as 
shown in (2), Fig. 47. 
A study of the figure 
will show that W is 
balanced by two equal 
parallel forces, each of 
which is equal to F. 
HenceTf=2^. The 
w mechanical advantage 
(j^) ^2) (2) is therefore 2. In (3) 

Fig. 48.— (1) One Movable and Two Fixed ^^SO the mechanical 
Pulleys. (2) Two Movable and Two Fixed advantao^e is 2, and the 
Pulleys. (3) Two Movable and Three . ? , , 

Fixed Pulleys. on\j gam secured by 



MACHINES 



87 



the use of the fixed pulley is one of direction, i.e. the effort 
may now act downward instead of upward as in (2). 

In (1), Fig. 48, one end of the rope is attached to the 
movable block, so that W is supported by three upward 
parallel forces, each of which is equal to F. Hence 
W='^F. A similar consideration of (2) will show that 
Tr= 4 F, and of (3), that W= 5 F. 

It is obvious from the cases already considered that 
whenever a continuous cord is used in the pulley system, 
the mechanical advantage is equal to the number of paral- 
lel forces acting against the resistance W. If n is the 
number of the parts of the rope supporting the mova- 
ble block, then n is the number of these parallel and equal 
forces of which TTis the sum. Therefore 

W = nF. (3) 

92. Principle of Work and the Pulley System. — Equa- 
tion (3) can be derived by applying the general law of 
work stated in § 89. If there are n portions of the rope 
supporting the movable pulley, and W is lifted 1 foot, 
for example, each portion of the cord must be shortened 
that amount. Consequently the effort 
F must move n feet. By the principle 
of work the product of the effort and 
the distance through which it acts 
equals the resistance multiplied by the 
distance through which it is moved ; or, 

W= nF. 

EXERCISES 

1. Diagram a set of pulleys by means of 
which an effort of 100 lb. can support a load of 
500 lb. 

2. What is the mechanical advantage of the 
system of pulleys shown in Fig. 49 ? Find the 
effort req^uired to balance a weight of 1200 lb, 




Fig. 49. — A Tackle. 



88 



A HIGH SCHOOL COURSE IN PHYSICS 



3. Each of two pulley blocks contains two sheaves. Show by a 
diagram how to arrange these into a system that will enable an effort 
of 75 Kg. to move a resistance of 300 Kg. 

4. Show by a diagram the best arrangement of two blocks, one 
containing two sheaves, the other containing one. Ascertain the 
mechanical advantage. 

5. In each case shown in Fig. 48 let the effort be applied to the 
movable block and the resistance W to the end of the rope in place of 
F. If the effort F moves 1 ft., how far will W be moved ? State the 
advantage secured in each instance. Note. — This plan is frequently 
employed in the operation of passenger elevators in tall buildings. 

6. In a pulley system consisting of a continuous cord attached at 
one end to a movable block containing one sheave, the rope passes 
through a fixed block having two sheaves. Find the effort required 
to support a block of marble weighing a ton. 




Fig. 50. — The Lever. 



3. THE PRINCIPLE OF THE LEVER 

93. The Lever. — Of all the simple machines the lever 
is the most common. It is of frequent occurrence in the 

structure of the skeleton of 
man and animals, as well as 
in many mechanical appli- 
ances. In its simplest form 
the lever is a rigid bar, as 
AB, Fig. 50, arranged to turn about a fixed point called 
the fulcrum. The effort is applied at the point J., and the 
resistance at B. 

Balance a meter stick upon a long wire nail, piercing it a little above 
the center, as shown in Fig. 

51. By means of a small A 45 cm mIScm B 

cord or thread, suspend a 
weight of 150 grams at 5, 15 
centimeters from the ful- 
crum 0. Now place a 
weight of 50 grams upon 
the opposite side of the 
fulcrum and move it along 
the bar until it just balances 




Fig. 51.— The Moment of the Effect F 
equals the Moment of the Resistance 
W. 



MACHINES 89 

the weight at jB. Call this point .1 . It will now be found that the 
distance yl O is 45 centimeters. From this it is plain that the force F 
multiplied by the arm AO is equal to the weight W multiplied by the 
arm BO. The experiment may be extended by using other forces and 
distances, but in every case it will be found that the product F x AO 
equals the product W x BO. 

The product of the applied force JF multiplied by its 
distance AO from the fulcrum is called the moment of 
that force (§ 45). Likewise, the product of the resist- 
ance W multiplied by its distance BO from the fulcrum 
is the moment of the resisting force. Thus the fact shown 
by the experiment may be stated as follows : 

The moment of the effort is equal to the moment of the 
resistance. 

The distances A and B are called the arms of the 
lever. Representing these distances by I and I' respec- 
tively, the law of the lever is represented by the equa- 
tion 

Fxl=Wxl'. (4) 

From this equation it is clear that the mechanical ad- 

W I 

vantage — (§ 90) is represented by the ratio -• Hence, a 

lever will support a iveight or other resistance — times as 
great as the effort. 

94. Classes of Levers. — Levers are usually divided into 
three classes (Fig. 52) depending upon the relative location 
of the fulcrum, the effort, and the resistance. 

(1) In levers of the first class the fulcrum is between 
the effort F and the resistance TF; as in the crowbar, beam 
balance, scissors, steelyard, pliers, wire cutters, etc. 

(2) Levers of the second class are distinguished by 
the fact that the resistance W is between the fulcrum 
and the effort JP; as in the nutcracker, wheelbarrow, etc. 



90 



A HIGH SCHOOL COURSE IN PHYSICS 



(3) Levers of the third class are distinguished by the 
fact that the effort F is between the fulcrum and the 
resistance W; as in the fire tongs, sheep shears, sewing- 
machine treadle, etc. 

^t o I'' 



^m 



I 'u I ' ' ' 





1st Class. 



Fig. 52. 



2d Class. 
■Levers of the Three Classes. 




If the forces acting on the lever are not parallel, it is 
spoken of as a bent lever. A hammer used in pulling a 

nail (Fig. 53) is a lever of this kind. 
Bent levers are frequently found in 
complicated machines ; as in farming 
implements, metal-working machin- 
ery, clocks, etc. 

In levers of all classes the arms are 
measured from the fulcrum on lines 
perpendicular to the lines which rep- 
resent the direction of the forces. 
The law stated in the preceding sec- 
tion holds for all cases. 

95. Extension of the Principle of 
Moments. — The relation of the mo- 
ments of the forces acting on the arms of a lever as ex- 
pressed in § 93 may be extended to cases in which any 
number of forces act upon the bar, provided they pro- 
duce equilibrium. The case may be illustrated by experi- 
ment as follows; 




Fig. 53. — The Hammer 
used as a Bent Lever. 



MACHINES 



91 



iT i I i I 




Let weights be placed on the balanced meter stick used in § 93 pre- 
cisely as shown in Fig. 54. The forces will be found to balance. 

Since there is no rotation of the lever, it is obvious that 
the forces tending to pro- 
duce a clockwise Q rota- 
tion are just balanced by 
those which tend to ro- 
tate the bar in the coun- 
ter-clockwise Q direc- ^^^" ^*' 
tion. On computing the moments of the forces, we find 
on the right-hand side 50 x 40 and 100 x 25. On the left 
we find 20 x 25 and 100 x 40. Noav if the moments acting 
clockwise be added, their sum will be found equal to the 
sum of those acting counter-clockwise; or, 

50 X 40 + 100 X 25 = 20 X 25 + 100 x 40. 
This equation illustrates a general law which may be 
stated thus: 

An equilibrium of forces results when the sum of the mo- 
ments which tend to make the lever rotate in one direction 
equals the sum of the moments which tend to make it rotate 
in the opposite direction. 

The mechanical advantage of the lever can also be found 

</ by applying the prin- 
ciple of work set 
forth in § 89. Let 
the lever shown in 
Fig. 55 turn slightly 
about the fulcrum 
until it has the position cd. Drawing the perpendiculars 
ac and hd^ we have from similar triangles 

ac : bd : : Oc : Od. 
But, since Oc equals OA and Od equals OB^ 

ac'.bd'. :AO:BO. 



w 

Fig. 55. — The Work Done by the Effort F 
Equals That Done upon the Weight W. 



92 



A HIGH SCHOOL COURSE IN PHYSICS 



Now the work done by the effort F is the product F x ac^ 
and the work done against the resistance TT is TF x hd. 
Since the work done by the agent is equal to that accom- 
plished by the lever, 



F X ac = W X bd, or W : F : : ac : bd : : AO : BO. 



(5) 



Therefore, the mechanical advantage of a lever is the ratio 
of the arm AO to which the effort is applied to the arm 
£0 on which the resistance acts. 



EXERCISES 

1. Show by diagrams the relative position of effort, resistance, and 
fulcrum in the following instruments of the lever type : oars, sugar 
tongs, lemon squeezer, rudder of a boat, pitchfork, spade, can opener, 
pump handle. 

2. Two boys weighing respectively 60 and 45 lb. balance on oppo- 
site ends of a board. If the fulcrum is 6 ft. from the larger boy, how 
far is it from the smaller one ? 

3. The arms of a lever of the first class are 3 ft. and 7 ft. What 
is the greatest weight that a force of 60 lb. can support? 

4. A lever of the second class is required to support a weight of 500 
kg. ; the effort to be applied is only 50 kg. If the bar is 20 ft. 
long, where should the weight be attached ? 

5. Two masses weighing respectively 10 and 15 kg. balance 
when placed at opposite ends of a bar 2 m. long. Where is the 
fulcrum ? 

6. The short arm of a lever is 30 cm. long, the long arm 270 cm. 

If the end of the long 
arm moves 1 cm., how far 
will the end of the short 
arm move? 

7. The forearm is 
raised by a shortening 
of the biceps muscle. 
Considering the forearm 
as a lever whose fulcrum 
is at the elbow, to what 
class does it belong? 




Fig. 56. — Wagon Scales. 



MACHINES 



93 



What is gained, force or speed? When the arm is being extended in 
striking a blow, to what lever class does it belong? 

8. The scales used in weigh- 
ing heavy loads, Fig. 56, are a 
series of levers arranged as shown. 
Explain how a small weight W 
can balance a load of coal weigh- 
ing a ton or more. 

9. Explain how a parcel is 
weighed bv means of the steel- 
yards shown in Fig. 57. Fig. 57. - Steelyards. 




4. THE PRINCIPLE OF THE WHEEL AND AXLE 

96. The Wheel and Axle. — A simple form of the wheel 
and axle is shown in Fig. 58. The applied force, or effort, 

acts tangentially to the rim of the 
wheel B through a cord, thus pro- 
ducing rotation. As the wheel 
revolves, the cord to which is at- 
tached the weight W is wound up 
around the axle A, and the weight 
is thus lifted. 

If the force F turns the wheel 
through one revolution, the dis- 
tance through which it acts is 
equal to the circumference of the 
wheel, or 2 irM, where M is the radius of the wheel. The 
work done by the agent is therefore F x 2 irR (§ 55). One 
revolution of the wheel lifts the weight W a distance 
equal to the circumference of the axle, i.e. to 2 7rr, where r 
is the radius of the axle. The work done therefore by 
the machine upon the weight is TT x 2 irr. Applying the 
principle of work as stated in § 89, 




Fig. 58. — The Wheel 
and Axle. 



F x2itR= Wxlirv', 
whence FxR = Wxr, orW:F::R:r. 



(6) 



94 



A HIGH SCHOOL COURSE IN PHYSICS 



The mechanical advantage of a wheel and axle is there- 
fore equal to the radius of the wheel divided by the radius 
of the axle. Hence, a wheel and axle may he used to over- 
come a resistance — times as great as the effort applied to 

the circumference of the wheel. 

It should be observed that the circumferences of the 
wheel and the axle bear the same ratio as their respective 
radii, and therefore may be used to replace R and r in 
equation (6). 

In the compound windlass, Fig. 59, two machines of the wheel and 

axle type are combined. The effort F acting along the circumference 

of the circle described by 
the crank produces a force 
that is transmitted by 
means of a small cog 
wheel, or pinion, to the 
rim of the second wheel 
to which the axle is at- 
tached. As the crank is 
turned, a rope is wound 
around the axle, thus lift- 
ing a weight or overcom- 
ing some other resistance. 
The mechanical advan- 
tage of a compound wind- 
lass can be calculated by 
resolving the machine into 
simple wheels and axles. 
The mechanical advan- 
tage of the compound 
machine is the product of 

the mechanical advantages of its constituent parts. 

In many instances in which the wheel and the axle are compounded 

the motion is transmitted by belts, chains, or cables, and occasionally 

by the friction of the circumferences. 

The wheel and axle and the pulley may be looked upon 
as special cases of the lever. Fig. 60 (1) shows that the 




Fig. 59. — A Compound Windlass, or Wiiich. 



MACHINES 



95 



^ 



Fx R = Wx r. 




^2:: 



effort F acting tangentially to the rim of the wheel really 
acts upon a lever arm equal to R^ the radius of the wheel. 
Similarly, the resist- 
ance W acts upon a 
lever arm equal to the 
radius of the axle. / 
Applying the principle \ 
of moments, 



(2) 



w 



(3) 



Fig. 60. 



The case of a single 
fixed pulley shown in 
(2) may be regarded as a lever of the first class whose ful- 
crum is at 0, or as a wheel and axle of which the radii are 
equal. Either view leads to the relation F=W. A sin- 
gle movable pulley shown in (3) is similar to a lever of 
the second class, the fulcrum being at 0, and the resistance 
at the center as shown. Again, applying the principle of 
moments, we find that TF= 2 ^ as in § 91. 



EXERCISES 

1. Show that the ordinary kitchen meat chopper and coffee 
grinder are examples of the wheel and axle. Give other commonplace 

examples. 

2. The radius of a wheel 
is 3 ft., and that of its axle 
12 in. What effort would be 
required to overcome a re- 
sistance of 600 lb.? 

3 . How nmch work would 
be done upon the resistance 
in moving it a distance of 10 
ft. ? How much work would 
have to be done upon the 

Fig. 61. — The Capstan. wheel? 

4. The arm of a capstan, Fig. 61, measured from the center, is 
2 m. : the radius of the barrel is 25 cm. What effort would be re- 




96 



A HIGH SCHOOL COURSE IN PHYSICS 



quired to produce a tension of 500 kg. in the rope attached to the 
barrel ? 

5. What is the mechanical advantage of the machines in Exercises 
2 and 4? 

6. If the effort were applied to the axle, and the resistance to the 
wheel, what would be gained by using a wheel and axle? Illustrate 
by using the wheel and axle described in Exer. 2. 

7. The pedal of a bicycle describes a circle whose radius is 7 
in. If the radius of the attached sprocket wheel is 3 in., find the pull 
on the chain when the foot pressure is 45 lb. 

8. If the front sprocket of a bicycle contains 21 teeth, and the rear 
one 7, how far will one turn of the pedal move a 28-inch wheel along 
the ground V Find the number of turns of the pedal per mile. 

9. The length of the crank 
arm shown in Fig. 62 is 10 in., 
and the radius of w is 6 in.; 
wheel w is belted to S, whose 
radius is 3 in.; and S is at- 
tached to Wi whose radius is 20 
in. One turn of the crank pro- 
duces how many revolutions of 

the wheel Wl A point on the rim of W moves how much faster than 

the crank? 

10. The large wheel of a sewing machine is 12 in. in diameter, 

and the small one to which it is belted is 3 in. One up-and-down 

movement of the treadle produces how many stitches? 




Fig. 62. 



5. THE INCLINED PLANE, SCREW, AND WEDGE 

97. The Inclined Plane. — Let AB, Fig. 63, represent 
an inclined plane whose surface is smooth and unbending. 
Let the height BO he h, and the length AB be I. If the 
effort F acting parallel to AB 
causes the ball whose weight is 
W to move the distance AB, the 
work done by the agent is F x I. 

The weight W is lifted a dis- 

tance equal to BO, and the work fig. 63. —The luciiued Plane. 




MACHINES 



97 




Fig. G4. — A Screw is a 
Spiral Incline. 



done is TF X li. According to the general principle of 

work, 

Fxl=Wxh. (7) 

W I 

The mechanical advantage — is therefore equal to -, i.e. 

F h 

the ratio of the length of the plane to its height. 

98. The Screw. — The screw is sometimes considered 

to be a modification of the inclined plane. If a right tri- 
angle, Fig. 64, be cut from paper and 

wound around a cylindrical rod, as 

shown, the hypotenuse of the triangle 

forms a spiral similar to the threads 

of a screw. The distance between two 

consecutive turns of the thread measured 

parallel to the axis of the rod is the 

pitch of the screw. 

The equation giving the mechanical advantage of the 

screw is readily derived by applying the principle of work 

(§ 89) as follows : 

Let the screw shown in Fig. 65 be turned once around 

by applying the effort F to the end of arm A^ tangent to 

the circle which it de- 
scribes. During one 
revolution the effort 
acts through the dis- 
tance 2 7rr, where r is 
the length of the arm. 
The work done is -F x 
2 irr. While the screw 
is making one revolu- 
FiG. 65. — The Screw. tion, the weight W is 

obviously lifted through a distance equal to the pitch of 

the screw, which may be represented by the letter s. The 

work done by the screw is therefore W x s. Hence 
8 



334 




98 



A HIGH SCHOOL COURSE IN PHYSICS 



Fx2'irxr = Wxs. 



(8) 




Fig. 66. — The 
Jackscrew. 



The mechanical advantage of the screw is therefore the 

ratio 2 irr/s. 

The screw as a mechanical power owes its importance 

to the fact that its mechanical advantage can be enor- 
mously increased simply by making r large 
and 8 small, as in the jackscrew, Fig. ^^^ 
used to lift buildings from their founda- 
tions. It is extremely useful in the form 
of bolts, wood screws, vises, clamps, presses, 
water taps, and in many other cases where 
a great multiplication of force is desired. 

99. The Wedge. — The wedge may be con- 
sidered as two inclined planes placed base to 

base. It is used for splitting logs (Fig. 67), raising heavy 

weights small distances, removing the covers from boxes, 

etc. The importance of the 

wedge is due to the fact that the 

energy imparted to it is delivered 

by the blows of a hammer or 

heavy mallet. Although the 

amount of friction to be overcome 

in driving a wedge is necessarily 

very large, by its use a man can 

overcome enormous resistances. Since friction cannot be 

disregarded, no definite relation between effort and re- 
sistance can be given. 

EXERCISES 

1. A ball weighing 10 lb. rests upon an inclined plane. If the 
height of the plane is 6 in. and the length is 30 in., what effort acting 
parallel to the plane will be required to hold the ball in equilibrium? 

2. On an icy slope of 45°, what force is required to haul a sled and 
load weighing a ton, neglecting friction? 

3. The radius of the wheel of a letter press is 12 in.; the pitch of 




Fig. 67. 



— One Use of the 
Wedge. 



MACHINES 99 

its screw, \ in. NeglectiDg friction, what pressure is produced by an 
effort of 50 1b.? 

4. Neglecting friction, what constant force must a team of horses 
exert in hauling a load of coal weighing 3000 lb. up an incline of 30°? 

5. What is gained by making an inclined plane of a given height 
longer? What is lost? Illustrate by means of examples. 

6. In a machine the effort of 50 lb. descends 20 ft., while a weight 
is raised 10 in. What is the weight? 

7. W^hat is the mechanical advantage of a screw press of which 
the pitch of the screw is 5 mm. and the diameter of the circle 
described by the effort 50 cm. ? 

8. A smooth railroad track rises 50 ft. to the mile. A car weigh- 
ing 20 T. would require how much force to keep it from moving down 
the slope? 

6. EFFICIENCY OF A MACHINE 

100. Friction and Efficiency. — On account of the fact 
that there is always a resistance due to friction wherever 
one part of a machine moves over another, some work 
must be done in moving the parts of the machine itself. 
The useful work done by a machine is therefore less than 
the work done upon it; i.e. W X d' is always less than 
F X d. The ratio of the useful work done hy a machine to 
the work done upon it is the efficiency of the machine. 

Example. — In the working of a pulley system an effort of 50 lb. 
acts through a distance of 20 ft. and lifts a weight of 180 lb. 5 ft. 
What is the efficiency of the system ? 

Solution. — The work done on the machine is 50 x 20, or 1000 
foot-pounds. The work done by the machine is 180 x 5, or 900 foot- 
pounds. The efficiency is therefore 900 -^ 1000, or 0.9. 

Efficiency is usually expressed as a percentage of the 
total work applied to a machine ; thus in the example 
above the efficiency is 90 %. 

101. Sliding and Rolling Friction. — Since friction 
always tends to decrease the efficiency of machinery, 
advantage is taken of every method that will reduce it 
to the smallest possible amount. 



100 A HIGH SCHOOL COURSE IN PHYSICS 

Let a block of wood or metal and a car of the same weight be 
drawn up a slight incline and the force measured by a dynamometer. 
It will be found that the car is more easily moved than the block. 

Again, place a piece of sheet rubber on the incline under the car. 
A greater force will be required to produce motion than before. 

The first experiment shows clearly a great difference 
between the sliding friction of the block and the rolling 
friction of wheels on a hard surface. The second experi- 
ment demonstrates the value of a hard, unyielding road 
when heavy loads are to be drawn. If, however, the 
wheels are provided with wide tires, they sink less deeply 
into the roadbed and meet with less resistance. The un- 
yielding surfaces of car wheels and the track enable a 
locomotive to pull enormous loads on account of the small 
amount of rolling friction. 

It is plain that in the axle bearings of ordinary vehicles 
the moving part must slide over the stationary part. The 
friction thus brought about is greatly re- 
duced by means of lubricating oil, but is 
avoided in the construction of bicycles, 
automobiles, etc., by substituting ball bear- 
ings, as shown in Fisf.GS. In this way the 

Fig. 68. - Ball ^ '. . ^ . . . p .-U Z^' 

Bearings of a moving part A3 IS separated trom the station- 
Bicycle Axle. ary part A by balls which roll as the wheel 

turns. Thus rolling friction takes the place of sliding 

friction. 

EXERCISES 

1. In what way are we dependent upon friction in the process of 
walking? Why do we encounter difficulty in walking on smooth ice? 

2. Why is it difficult for a locomotive to start a train on wet rails ? 
How is the difficulty overcome? 

3. Why do smooth nails hold two pieces of wood together ? State 
other ways in which we take advantage of friction. 

4. Calculate the efficiency of a wheel and axle when an effort of 
20 lb. acting through 30 ft. lifts a weight of 80 lb. 7 ft. 




MACHINES 101 

5. On account of the loss of energy due to friction in a pulley sys- 
tem, an effort of 70 kg. acting through 30 m. moves a resistance of 
340 kg. through 5.5 m. What is the efficiency of the machine? 

6. What is the efficiency of a screw if an effort of 5 kg. applied at 
the end of an arm 1 m. long produces a pressure of 4000 kg., the pitch 
of the screw being 4 mm. ? 

7. The efficiency of an inclined plane is 50%. If the length of the 
plane is 20 ft. and its height 4 ft., what effort acting parallel to the 
plane will be required to move a body weighing 500 lb. ? 

SUMMARY 

1. The simple machines are the pulley, lever, wheel 
and axle, incline plane, screw, and wedge. Complicated 
machinery is made up of simple machines (§ 88). 

2. The work done by an agent upon a machine is equal 
to the work accomplished by it. This is known as the 
Principle of Work. Hence a machine may gain force at 
the expense of distance (or speed), or it may gain distance 
(or speed) at the expense of force (§ 89). 

3. The mechanical advantage of a machine is the ratio 
of the resistance overcome by it to the effort applied to it; 
ix.WiF (§90). 

4. When a continuous cord is used in a system of pul- 
leys, the equation \^W — nF^ in which n is the number of 
parts of the rope supporting the movable block (§ 91). 

5. When a lever is used, the moment of the effort 
equals the moment of the resistance, ov F xl— W xV (^^ 93). 

6. Levers are classified according to the relative po- 
sition of fulcrum, effort, and resistance. Lever arms are 
always measured from the fulcrum on lines perpendicular 
to the acting forces (§ 94). 

7. When several forces act at different points on a 
lever, the sum of the moments tending to produce a clock- 
wise rotation equals the sum of the moments tending to 
produce rotation in the opposite direction (§ 95). 



102 A HIGH SCHOOL COURSE IN PHYSICS 

8. The equation of the wheel and axle is F x M = 
TF X r (§ 96). 

9. The equation of the inclined plane is F x I = 
TrxA(§97). 

10. The equation of the screw is F x 27rr = W 
X s (§ 98). 

11. Friction tends to reduce the work accomplished by 
a machine. The efficiency of a machine is the ratio of 
the useful work done by a machine to the work done 
upon it (§ 100). 

12. In general the friction of sliding parts of machinery 
is greater than that of rolling parts. Friction may be re- 
duced by lubricants, or by substituting rolling friction 
for sliding friction, as in the case of ball and roller 
bearings (§ 101). 



CHAPTER VII 



MECHANICS OF LIQUIDS 



1. FORCES DUE TO THE WEIGHT OF A LIQUID 

102. Pressure of Liquids against Surfaces. — When a 
hollow rubber ball or a piece of light wood is forced under 
water, the body resists the action of the force submerging 
it and manifests a strong tendency to return to the surface. 
The fact that some bodies float on water and other liquids 
shows also that there exists a force acting against the 
lower surfaces sufficient to counteract their weight. 

Let a lamp chimney against the end of which a card has been 
placed be forced partly under water, as shown in Fig. 69. The card 
will be pressed firmly against the end of the 
chimney by a force acting in an upward di- 
rection, and a large quantity of shot or sand 
may be poured into the vessel before the card 
is set free. The person performing the ex- 
periment will also perceive that a strong 
upward force acts against the hand. By 
lowering the chimney to a greater depth, the 
upward force against the hand will be in- 
creased and a larger quantity of shot or sand 
will be required to free the card. 

103. Relation between Force and 
Depth. — The manner in which the 
force exerted by a liquid against a surface varies with the 
depth may be experimentally tested 'by means of a gauge 
constructed as shown in Fig. 70. 

A "thistle tube" having a stem about 80 centimeters long is 
bent at an angle of 90 degrees about 35 centimeters from one end. 

103 




Fig, 69. — A Liquid Exerts 
an Upward Pressure 
against the Vertical 
Tube. 



104 



A HIGH SCHOOL COURSE IN PHYSICS 



II 




A piece of thin sheet rubber is tied tightly over the large end A. If a 
drop of ink is placed in the tube at B, its movements along the tube 

will indicate changes in the force against the 
rubber surface A. Furthermore, the dis- 
tance the drop moves vv^hen A is submerged 
in a liquid will be very nearly proportional 
to the applied force. 

Let the gauge be clamped in the position 
shown in the figure, and let a graduated lin- 
ear scale be attached to the horizontal tube 
containing the drop of ink. Now let the po- 
sition of the drop be read upon the scale and 
a tall vessel of water brought up until the sur- 
face J. is 3 centimeters beneath the free sur- 

Pjq 70 -pjjg Upward ^^^® °^ ^^® liquid. Let the new position of 

Force against A Varies the drop be read and its displacement com- 

with the Depth. puted. If, now, the positions of the drop be 

read after submerging the surface A successively to the depths of 6, 9, 

and 12 centimeters, the movements of the drop will be found to be 

proportional to the depths. 

Therefore, as shown by this experiment, the upward 
force exerted hy a liquid of uniform density against a given 
surface is directly proportional to the depth of that surface. 

104. Direction of Forces at a Given Depth. — Common 
experience shows that liquids press against surfaces that 
are vertical or oblique as well as horizontal. In every 
case the force' exerted is perpendicular to the surface 
against which it acts. Thus Avater will be forced through 
a hole in the side of a pail near the 
bottom as well as through one in the 
horizontal bottom itself. A compari- 
son of the forces in all directions at 
a given depth may be made by modi- 
fying slightly the construction of the 
gauge shown in Fig. 71 as follows : 

^,,, , ,, i.-i J.-. Fig. 71. — Forces Acting at 

Cut the glass tube about 1 centimeter ^ p^j^^ ^^e the Same in 

from the bulb, and insert a piece of rubber all Directions. 




MECHANICS OF LIQUIDS 105 

tubing about 10 centimeters long. Now, if the bulb is lowered in a 
large vessel of water, the position of the drop will not change when 
the rubber surface is turned in different directions, provided the depth 
of its center is kept constant. 

This experiment, together with the one described in 
§ 103, leads to the following conclusions : 

(1) The force exerted hy a liquid at a given depth is the 
same in all directiojis^ aiid 

(2) The force exerted hy a liquid in any direction is 
directly proportional to the depth. 

105. Pressures Due to the Weight of a Liquid. — From 

a study of Fig. 72 it may readily be observed that the 
forces exerted by liquids against surfaces «^ ^ 

are due to the weight of the liquid, i.e, to 
gravity. If a vessel were filled with 
smooth blocks of wood of equal size and 
density, block 2 would suffer a downward 
force equal to the weight of block i, and 
would in turn exert an equal and opposite fig. 72. — Forces 
reaction upward against it. Likewise, 3 Due to Gravity. 
must support the weight of 1 and ^, 4 must support i, ^, 
and 5, etc. In each instance the upward reaction is equal 
to the downward action. Thus it is clear that at a given 
surface the upward and downward forces are equal, and 
also that the force against any horizontal surface is pro- 
portional to the depth of that surface below the upper 
surface of the blocks in the vessel. 

Now liquids, unlike solids, do not tend to keep their 
form, but must be supported on the sides. Hence, if we 
imagine the vessel in Fig. 72 to contain a liquid, section 
4, for example, will press sidewise against 8 and i^, whose 
reactions back against 4 preserve equilibrium. If the 
liquid were without weight, section 4, for instance, would 
suffer no crushing force due to the weight of the portions 



9 


1 


5 


10 


2 


6 


11 


3 


7 


12 


4 


8 



106 



A HIGH SCHOOL COURSE IN PHYSICS 










/! 


:- - : 


D 


;==^ 


is^ 


t^ 



above it, and consequently would exert no lateral force 
against the portions surrounding it. 

106. Total Pressure on a Given Area. — If the experiment 
described in § 102 be repeated, and water substituted for 

the shot or sand 
(Fig. 73), it will 
be found that the 
card is set free 
from the end of 
the chimney at the 
instant the water 
on the inside 
reaches the height 
of that on the out- 
side. 

If the chimney is 
cylindrical, the 
downward force 
within the chimney is obviously equal to the weight of the 
column of water ABCD. If the area of the end of the 
chimney is a square centimeters, and the depth h centi- 
meters, the volume of the column of water in the chimney 
is ah cubic centimeters. Since the density of water is 1 
gram per cubic centimeter (§ 9), the weight of the column 
ABCD is ah grams. Hence the force exerted on the card 
is ah grams. If another liquid is used whose density is d 
grams per cubic centimeter, the weight of the column, and 
hence the force, is ahd grams. 

The following rule may therefore be given: 

The force exerted hy a liquid on any horizontal surface is 
equal to the weight of a column of the liquid whose base is 
the area pressed upon^ and whose height is the depth of this 
area below the surface of the liquid^ or Force — ahd. 



(s) 

Fig. 73. — Force against 5C is Equal to the 
Weight of Column ABCD. 



MECHANICS OF LIQUIDS , 107 

Since the force exerted by a liquid against a surface at 
a given depth is the same in all directions (§ 104), the 
following rule for computing the force against surfaces 
that are not horizontal is often employed ; 

The force exerted hy a liquid against any immersed surface 
is equal to the weight of a columii of the liquid whose base is 
the area pressed upon^ and whose height is the distance of the 
center of mass of this area below the surface of the liquid. 

In the English system the area should be expressed in 
square feet, the depth in feet, and the density of the liquid 
in pounds per cubic foot. The density of water may be 
taken as 62.5 pounds per cubic foot. 

Example. — A tank 4 ft. deep and 8 ft. square is filled with water. 
Find the force exerted against the bottom and one side. 

Solution. — The area of the bottom of the tank is 8 x 8, or 64 sq. 
ft. Hence the volume of the column whose weight equals the force 
exerted on the bottom is 4 x 64, or 256 cu. ft. The force against the 
bottom of the tank is therefore 256 x 62.5 lb., or 16,000 lb. 

The area of one side of the tank is 4 x 8, or 32 sq. ft. The depth 
of the center of mass of the side is 2 ft. Hence the volume of the 
column of water whose weight equals the force exerted against the 
side is 2 X 32, or 64 cu. ft. The force exerted by the water upon this 
side is therefore 64 x 62.5 lb., or 4000 lb. 

The term pressure should be confined to the meaning of 
force per unit area. The pressure at a given point is ex- 
pressed in terms of the force exerted over a unit area at 
that depth ; for example, 1000 grams per square centi- 
meter, 15 pounds per square inch, etc. 

EXERCISES 

1. Find the entire force exerted against the bottom of a rectangu- 
lar vessel 5x8 cm. and' filled with water to a depth of 15 cm. 

2. Find the pressure per square foot at the bottom of a pond 10 
ft. in depth. 

3. A cylindrical glass jar 5 cm. in diameter is filled to a depth of 
15 cm. with mercury. Find the force against the bottom and tlie 



108 A HIGH SCHOOL COURSE IN PHYSICS 

pressure per unit area. The density of mercury is 13.6 g. per cubic 
centimeter. 

4. A tank is 4 ft. wide, 8 ft. long, and 3 ft. deep. Compute 
the force exerted against one end and the bottom when the tank is 
full of water. 

5. At a depth of 10 m. of sea water, what is the pressure in grams 
per square centimeter? (The density of sea water is 1.026 g. per cubic 
centimeter.) 

6. At a depth of 25 ft. of sea water, what is the pressure per 
square inch ? 

Suggestion. — Find first the force exerted on a surface 1 ft. square 
at the given depth. 

7. A cubic inch of mercury weighs 0.49 lb. Compute the force 
exerted against the bottom and one side of a glass tank 4 in. wide, 6 
in. long, and .5 in. deep when full of mercury. 

8. Find the force exerted against the bottom of a cubical vessel 
whose volume is 1 liter when the vessel is filled with mercury. 

9. What depth of water will produce a pressure of 1 lb. per square 
inch? 

10. What is the pressure per square centimeter at the bottom of a 
column of mercury 76 cm. in height? (For the density of mercury 
see Exer. 3.) 

11. To what height would a mercurial column be supported by a 
pressure of 1000 g. per square centimeter? 

12. A gauge connected with the water mains of a city showed a 
pressure of 65 lb. per square inch. What was the height of the water 
in the standpipe above the level of the gauge ? 

Suggestion. — Find the depth of water required to produce a pres- 
sure of 65 lb. on a surface of 1 sq. in. 

13. A diver is working at a depth of 45 ft. How much is the 
pressure per square inch upon the surface of his body ? 

14. A rectangular block of wood is placed under water so that its 
upper face, which is 8 x 10 cm., is 20 cm. below the surface. If the 
thickness of the block is 4 cm., what is the force exerted by the liquid 
against each of its faces? 

15. How much is the force against a dam 20 ft. long and 10 ft. high 
when the water rises to its top ? 

16. A hole in the bottom of a ship which draws 30 ft. of water is 
temporarily covered with a piece of canvas. How much is the pres- 
sure against the canvas from the outside? 



MECHANICS OF LIQUIDS 



109 



17. The water level is at the top of a dam 30 ft. high. Compute 
the pressm^e per square foot at the bottom of the dam. How much is 
the pressure halfway dowu ? 

18. If the dam in Exer. 17 is 100 ft. long, how much is the total 
force against its surface ? 

107. Pressure in Vessels of Different Shapes. — It may 

be correctly inferred from § 106 that the force exerted by 
a liquid against a given area does not depend on the shape 
of the vessel containing the liquid used, inasmuch as the 
computation of this force involves only the area and depth 
of the surface pressed upon and the density of the liquid. 
This fact can be demonstrated experimentally as follows : 

Let a glass funnel be selected the mouth of which is of the same area 
as the end of the lamp chimney used in § 106. Place a card across the 
mouth of the funnel and submerge it, as shown 
in Fig. 74. The card will be pressed against 
the funnel with the same force as it was when 
the chimney was used, i.e. with a force equal 
to the weight of a column of water ABCD. 
If water is now poured into the stem of the 
funnel at E, the card will become free precisely 
when the level of the water in the funnel has 
reached the height of the water in the vessel 
outside. In this condition the water in the 
funnel exerts the same force downward against 
the card as the water on the outside exerts 

in an upward direction. Hence the downward force is equal to the 
weight of the column of water ABCD. In other words, the force 
downivard against the card is exactly the same as it was when the cylindri- 
cal chimney was used. 

108. The Hydrostatic Paradox. — An apparent contra- 
diction arises when we 
apply the laws of liquid 
pressure to vessels of 
the forms shown in Fig. 

a) ^ ^ (2) ^ ^ (3) ^ '^^* L^^ ^^® vessels have 

Fig. 75. — The Hydrostatic Paradox. bases of equal size and 




Fig. 74. —Equal Down- 
ward Forces in Ves- 
sels of Different 
Shapes. 




no 



A HIGH SCHOOL COURSE IN PHYSICS 



be filled to the same depth with water. In each case the 
force exerted by the water against the bottom is equal to 
the weight of the liquid column ABQB. Because it is 
apparently an impossibility for different masses of a liquid 
to produce the same pressure, this conclusion is often 
called the hydrostatic paradox. 

An application of the laws of pressure in liquids to vessel (3), Fig. 
75, will show how the total pressure on the bottom BC can be far 
greater than the weight of liquid contained in the vessel. Although 
the total pressure on the surface i^C is equal to the weight of a column 
A BCD of the liquid (§ 107), there are upward forces against the 
surfaces e/and gh equal to the weight of the liquid that would be re- 
quired to fill the spaces Aefa and bghD. Hence the resultant of all 
the forces is the difference between the downward force on i^C and 
the upward forces on ef and gh. This is obviously the weight of the 
liquid in the vessel. 

109. A Liquid in Communicating Vessels. — Let tubes 
of various shapes and sizes open into a hollow connecting 

arm, as shown in Fig. 76. Any 
liquid poured into one of the tubes 
will come to rest at the same level 
in all. Although different quan- 
tities of the liquid are present in 
the several tubes, yet for the same 
depth the parts are in equilibrium. 
Fig. 76. — Liquid Level in The explanation is as follows : 

Communicating Vessels. . . . 

Let two vessels containing water be in 

communication, as shown in Fig. 77. Let A be the area of a cross- 
section of the connecting tube. The force tend- 
ing to move the water to the right is Ahd, 
where h is the depth of the center of the area 
considered, and d the density of the water. 
The force tending to move the liquid to the left 
is Ah'd. These two forces will be in equilibrium 
only when h and h' are equal, i.e. when the 
upper surfaces in the two vessels lie in the same fiq. 77. _ a Liquid in 
horizontal plane. Equilibrium. 





MECHANICS OF LIQUIDS 



111 



EXERCISES 

1. A cone-shaped vase has a base of 100 cm.^ and is filled with 
water to a depth of 45 cm. Find the force and pressure per square 
centimeter acting on the bottom. 

2. The water in a reservoir supplying a city is 150 ft. above an 
opening made in a pipe being laid along a street. Find the pressure 
in pounds per square inch required to prevent the water from running 
out. Ans. 65.1 lb. 

3. A glass tube 1 m. long is filled with mercury (density 13.6 g. 
per cubic centimeter). Find the pressure against the closed end of 
the tube in grams per square centimeter when the tube is (1) vertical 
and (2) inclined at an angle of 45°. 

Ans. 1360 g. per square centimeter. 
961.7 g. per square centimeter. 

4. A column of water is lifted 25 ft. in a pipe. Calculate the 
pressure per square inch that it exerts against the bottom of the pipe. 



2. FORCE TRANSMITTED BY A LIQUID 




-The Multiplication of Force by a Liquid. 

110. Transmission of Pressure — PascaPs Law. — Let a 

vessel of the form shown in (1), Fig. 78, be filled with 
water to the point a. A pressure will be exerted on every 



112 



A HIGH SCHOOL COURSE IN PHYSICS 



square centimeter of area depending on the depth of that 
area. The force exerted upward against the shaded area 
AB, assumed to be 100 square centimeters, is 100 h grams, 
if h is the depth of the water in the tube. This force is 
entirely independent of the area of the portion of the 
vessel at a. Let this area be 1 square centimeter. Now, 
if 1 cubic centimeter of water is poured into the vessel, 
the depth of the liquid is increased 1 centimeter, and the 
depth of the surface AB becomes h + 1 centimeters. The 
force now exerted against AB is 100 (A + 1) grams, 
i.e. each square centimeter of AB receives an additional 
force of 1 gram. Hence, the force exerted on a unit area at 
a is transmitted to every unit area within the vessel. 

This fact was first published in 1663 by Pascal, a 
French mathematician. The law may be expressed as 
follows : 

Force applied to any area of a confined liquid is trans- 
mitted undiminished by the liquid to every equal area of the 
interior of the containing vessel and to every part of the 
liquid. 

111. The Hydraulic Press. — In the discussion of Pas- 
cal's Law given in the preced- 
ing section, it makes no dif- 
ference whether the pressure 
added to the small area a is 
produced by a gram of water 
poured into the tube or ex- 
erted by a small piston fitting 
the tube, as shown in (2), 
Fig. 78. The shaded area AB 
to which the force is trans- 
mitted by the liquid may be 
Fig. 79. — Au Hydraulic Press, made the area of a large piston. 




MECHANICS OF LIQUIDS 



113 



as shown. In this case a force of 1 gram on the small 
piston will be transmitted to each square centimeter 
of the large one. Hence 1 gram on the small piston will 
balance 100 grams placed on the large one. Furthermore, 
any effort F applied to the small piston will balance a re- 
sistance 100 times as great as itself. Hence a mechanical 
advantage (§ 90) of 100 is secured. The hydraulic press, 
a machine employed in factories for exerting great force, 
is one of the most important applications of Pascal's Law. 
(See Fig. 79.) 

The hydraulic press is analyzed in Fig. 80. When the lever L is 
raised, the small piston P is lifted, and water from the cistern T 
enters the cylinder B through 
the valve v. As the small pis- 
ton is forced down, v closes, 
and the water in the cylinder 
B is driven past the valve v' 
into the chamber below the 
large piston P'. By Pascal's 
Law, the force exerted against 
the large piston P' is as many 
times that applied to P as 
the area of P' is times that 
of P. In other words, the 
mechanical advantage is 
P' /P. By making P very 
small and P' large, any de- 
sired mechanical advantage 
may be secured. Thus hand 
presses are sometimes used 




Fig. 80. 



— Sectional Diagram of the 
Hydraulic Press. 



that are capable of exerting a force of several hundred tons. 

112. Principle of Work and the Hydraulic Press. — It 

may be observed from (2), Fig. 78, that when the effort F 
moves the small piston a distance of 1 centimeter, the 
large piston, having to make room for the 1 cubic cen- 
timeter forced below it, must rise ^^^ of a centimeter. 



114 A HIGH SCHOOL COURSE IN PHYSICS 

Although the larger force is 100 times as great as the 
smaller, it acts through only ^^^ as great a distance. 
Hence the work done by the larger piston as it rises is 
no greater than the work done on the smaller. 

113. Artesian or Flowing Wells. — The tendency of 
water to flow from a point of higher level to one of lower 
level has a wide application in artesian, or flowing, wells. 
In many localities there are formed so-called artesian 
basins of great extent in which a stratum of porous ma- 
terial, as sand or other substance through which water 
can pass with comparative ease, lies between strata of 
clay or rock which are impervious to water. At distant 
points these layers have been crowded to the surface 
by geologic processes where the porous layer has^ been 
laid bare, thus rendering the entrance of water pos- 
sible. Hence when borings are made into the earth 
through the various layers and into the porous stratum, 
water often rises to the surface when it is lower 
than the region where the water enters the porous 
layer. 

Deep artesian wells exist at St. Louis, Mo. ; Columbus, 
Ohio ; Pittsburg, Pa. ; and Galveston, Texas. Noted 
wells are found at Passy, France (1923 feet); Berlin, 
Germany (4194 feet); Leipzig, Germany (5735 feet). 

114. Supplying Cities with Water. — Comparatively few 
cities are so favorably situated that they derive their 
supply of water from mountain springs. The output of 
such springs is collected in large artificial reservoirs or 
lakes from which it is piped to the towns where it is dis- 
tributed in the usual manner. In such cases the elevation 
of the reservoir is such as to produce an adequate pressure 
for all ordinary purposes. The pressure may be estimated 
at 43.5 pounds per square inch for every 100 feet of 
elevation. 



MECHANICS OF LIQUIDS 



115 



In towns, however, whose location is less favored by 
nature, the water from springs or wells must be elevated 
by means of pumps (§ 156) to suitable reservoirs or tanks 
constructed upon high ground from which it is distrib- 
uted through pipes to the consumers. In some large 
cities no reservoir is used, the pumping engines being so 
adjusted as to supply the water at a given pressure as 
fast as it is consumed. 

115. Water Motors. — In cities where water is delivered 
through pipes under sufficient pressure, its power may 
be utilized in running sewing ma- 
chines, polishers, lathes, etc., by ..^^^ ^Nr-^*" 
employing a rotary water motor. 
A common type is shown diagram- 
matically in Fig. 81. Water is- 
sues with great velocity from the 
jet J against cup-shaped fans at- ,^_,_^^___ 
tached to the axle of the motor. j^ 
These are inclosed in a metal case fig. 81. — Sectional View of a 
O from which the water flows into Water Motor. 
the sewer or other drain. The impact of the water against 

the fans suffices to turn 
the shaft to which are 
connected either di- 
rectly or by belts the 
machines that it is de- 
sired to operate. 

116. Water Wheels. 
— An elevated body of 
water, like a lifted 
weight, is a source of 

Fig. 82. — An Oversliot Water Wheel. . <• i t. • 

potential energy. It is 
the falling of this water to a lower level that supplies the 
power for operating innumerable mills, factories, electric 





116 



A HIGH SCHOOL COURSE IN PHYSICS 



power plants, etc., found throughout this and other coun- 
tries. In order to enable water to do the required work, 
the so-called overshot water wheel has long been employed. 
A wheel of this kind is shown in Fig. 82. It is plain that 
the impact and weight of the moving water from above 
the wheel conspire to turn the wheel, to which is geared 
or beltea the machinery to be operated. Another old but 
common form of water wheel is the undershot wheel. In 
this case the wheel is turned by the impact of the current 
of water against fans at the bottom. 

The most efficient form of water wheel, however, is the 
turbine^ which is utilized in all modern plants employing 

water power. Water 
is conducted from the 
reservoir above a dam 
through a closed cylin- 
drical tube, or flume, 
F^ Fig. 83, to a penstock 
(shown by the dotted 
lines) which surrounds 
the stationary iron case 
T containing the rotat- 
ing wheel, or turbine. 
This case rests upon the 
floor of the penstock 
-and is submerged in water to a depth equal to the " head," 
or height,^ of the water supply. The turbine is attached 
to the shaft S and is set in rotation when the water is ad- 
mitted. The small shaft P serves to control the size of the 
openings in the case through which the water gains entrance 
to the turbine. Figure 84 is a sectional view through the 
turbine R and the case aS'. Water enters as shown by the 
arrows and strikes the blades of the turbine at the most 
effective angle for producing rotation. When the water 




Fig. 83. — A Water Turbine. 



MECHANICS OF LIQUIDS 



117 



has expended its energy, it falls from the bottom of the 
case into the tailrace below the penstock. The efficiency 
(§ 100) of turbines is frequently 
as high as 85 or 90 per cent. 

117. Hydraulic Elevators. — 
Another use made of the energy 
of water under pressure is in the 
operation of elevators, or lifts. 
One form is shown in Fig. 85. 
When water is admitted to the 
cylinder (7 through the control- 
ling valve F^ the piston is forced 
to the right, thus producing a 
pull upon the cable (shown by the dotted lines) and caus- 




FlG. 



84. — Section of a Water 
Turbine. 



III II 1 1 r I.J ' I ■ > 



l| II II 



'I' II I' 




" ^WWWWW^^WNNJO^ 



w 



ing the car A to ascend. Check- 
ing the flow of water by means of 
the rope r stops the car, while 
turning valve V to the position 
shown in the figure allows the 
water to flow from the cylinder 
and causes the car to descend by 
its own weight. A study of the 
figure will show that the car moves 
four times as fast and four times 
as far as the piston. 

In another form the piston is a 
long vertical cylinder extending 

from the bot- 
tom of the 
car into a 
deep hollow 

Fig. 85. — Au Hydraulic Elevator. Cvlinder set 

in the ground. The admission of water under pressure 
acting against the lower end of the long piston lifts the 




118 



A HIGH SCHOOL COURSE IN PflYSICS 




car. The method of controlling such elevators is the same 

as that shown in Fig. 85. 

118. The Hydraulic Ram. — The hydraulic ram is a 

useful device for automatically elevating water in small 

quantities when an 
abundant supply of 
spring water is available 
which has a fall of only 
a few feet. Water flows 
from the source through 

Fig. 86. — The Hydraulic Kam. the pipe P, Fig. 86, and 

out through the opening at v. As the water increases in 
speed, the valve at v is lifted, which causes a sudden inter- 
ruption of the flow in P. Since the momentum of the 
water cannot be destroyed instantaneously, a portion of the 
water is driven forcibly past valve v' into the air chamber 
0. A slight rebound of the water in the large pipe re- 
lieves for a moment the pressure against valve v, which 
falls by its own weight, thus opening again the orifice at 
that point. A repetition of the process causes more water 
to enter Q until finally the pressure is sufficient to lift a 
portion of the water to a height of many feet. The com-^ 
pressed air in O acts as an elastic cushion and serves also 
to keep a steady flow of water in the small pipe. The 
quantity of water delivered by an hydraulic ram is de- 
pendent on the height of the source, the elevation to 
which it is to be lifted, the friction of the pipes, the 
length of pipe P, and the size of the ram itself. 



EXERCISES 

1. The area of the small piston of an hydraulic press is 2 cm.* 
and that of the large one 80 cm.^ How much force will 50 Kg. 

applied to the former produce upon the latter ? 

2. The small piston of an hydraulic press is operated by a lever of 
the second class 4 ft. in length, and the piston rod is attached 12 in. 



MECHANICS OF LIQUIDS 119 

from the fulcrum. If the diameters of the pistons are 1 in. and 8 in. 
respectively, how great an effort will produce a force of 2 T. ? 

3. If the effort applied to the small piston in Exer. 2 moves through 
1 ft., how much will the large piston be raised ? 

4. A piston moves in a cylinder that is in communication with a 
water system whose pressure is 65 lb. per square inch. If a force of 
1 T. is to be developed by the piston, what is the least diameter that 
it can have? Ans. 6.26 in. 

5. Pressure against a piston 20 cm. in diameter is produced by a 
column of water 30 m. high. Calculate the force against the piston 
and the work performed when the piston moves 4 m. 

6. Give suitable dimensions to the pistons and lever of an hydraulic 
press in order that an effort of 1 lb. may produce a force of 3000 lb. 

. ARCHIMEDES' PRINCIPLE 

119. Buoyancy of Liquids. — It is a matter of common 
observation that bodies apparently become lighter when 
placed under water. If the hand, for example, be sub- 
merged in a vessel of water, it becomes evident at once 
that it is supported almost without muscular effort. Let 
a one- or two-pound stone be weighed in air and then 
weighed again while immersed in water. A decrease of 
several ounces will be observed. When blocks of wood or 
many other bodies are placed in water, the buoyancy of 
the water is sufficient to cause them to float. 

120. The Principle of Archimedes.^ — On account of the 
importance of the laws relating to the apparent decrease 

1 Archimedes (287-212 B.C.). The name of Archimedes will be re- 
membered by students of Physics in connection with the buoyant action 
of liquids on immersed solids. The story is related that Hiero, King of 
Syracuse, had ordered a crown of pure gold which, when delivered, al- 
though it was of the proper weight, was suspected to contain a quantity of 
silver. Archimedes was asked to investigate. A method of procedure 
occurred to him while in the public bath as he noticed that his body ex- 
perienced a greater buoyancy the more completely it was submerged. 
Recognizing in this effect the key to the solution of the problem, he 
leaped from the bath and hurried homeward, exclaiming "Eureka! 



120 



A HIGH SCHOOL COURSE IN PHYSICS 



in weight that bodies undergo when submerged in a liquid 
three experiments will be described: 

1. Measure the dimensions of a metal cylinder or rectangular 
metal block, and compute its volume in cubic centimeters. From the 

volume compute the weight of an 
equal volume of water. Suspend 
the metal body from one arm of a 
balance, and ascertain its weight in 
air. Weigh* the body also when sub- 
merged in water, and compute the 
loss of weight. Compare the loss 
of weight with the weight of an 
equal volume of water found at first. 
2. From one arm of a balance, 
Fig. 87, suspend a metal cylinder A 
and the bucket B whose capacity is 
precisely equal to the volume of 
the cylinder. Counterbalance these 
by means of sand or weights placed 
in the opposite scale pan. Submerge A in water, and equilibrium will 
be destroyed. Fill the bucket with water, and equilibrium will be 
restored to the system. 

3. Place a small glass beaker (or tumbler) on one pan of a balance, 
and suspend a stone from beneath the same pan. Counterpoise by 




Fig. 87. 



Verifying Archimedes' 
Principle. 



Eureka!" which means "I have found it!" Experiments showed 
that equal masses of gold and silver weigh unequal amounts when sub- 
merged in water. Therefore, when the crown was weighed against an 
equal mass of pure gold, both being submerged, the fraud was at once 
detected. 

Archimedes is regarded as the founder of the science of Mechanics. 
From his time nearly 2000 years elapsed before any great advance was 
made. His investigation of levers is noteworthy. He is said to have made 
this remark, " Give me a fulcrum on which to rest my lever and I will 
move the earth." Among his countrymen, he was probably best known 
on account of the numerous instruments of war which he invented. 

As a mathematician, Archimedes was first to determine the value of tt, 
and the first to compute the area of a circle. He is supposed to have been 
killed by a Roman soldier while engaged in the investigation of a problem 
in geometry. 



MECHANICS OF LJQUIDS 



121 



using sand, shot, or weights. Next fill a vessel provided with a spout 
with water, and allow all the excess to flow out at the spout. Place an 
empty tumbler under the spout, and lower the counterpoised stone 
into the water. Completely submerge the stone, and catch all the 
displaced water, the volume of which will be equal to that of the 
stone. Pour the displaced water into the beaker on the scale pan, and 
equilibrium will be restored. 

From the results obtained by making any one of the 
experiments just described Archimedes* Principle may 
be deduced: 

A body immersed in a liquid is buoyed up by a force equal 
to the weight of the liquid that it displaces. 

121. Explanation of Archimedes' Principle. — The rea- 
sons for Archimedes' law become clear when the laws of 
liquid pressure stated in § 104 are applied 
to a submerged body. Let a rectangular 
block abed be immersed in a liquid, as 
shown in Fig. 88. The force against the 
upper surface of the block is the weight 
of the column of the liquid efad and acts 
downward. The force exerted by the 
liquid against the lower surface of the 
block cb is the weis^ht of a column of the ^^^ 
liquid efbc and acts upward. The re- 
sultant of these two forces is obviously 
in an upward direction and equal to the weight of a volume 
abed of the liquid. The lateral forces exerted by the liquid 
against any two opposite faces of the block are equal and 
opposite, and therefore produce no tendency to move the 
block. Hence the immersed body is buoyed up by a force 
equal to the weight of a volume of the liquid equal to its 
own volume, i.e. to its displacement. It should be ob- 
served that the depth to which the body is submerged does 
not affect the final result. 




!. — The Up- 
ward Force Ex- 
ceeds the Down- 
ward Force. 



122 



A HIGH SCHOOL COURSE IN PHYSICS 



(ij 




'^ \- m\ |fe 



If the weight of the body submerged in the liquid just 
equals the weight of the displaced liquid, the body will be 
in equilibrium and remain where it is placed provided the 
density of the medium be uniform. If the body weighs 
more than the amount of liquid displaced, it will sink; if 
less, it will rise to the surface and float. 

122. Floating Bodies. — l. Obtain a square stick of pine about 
30 centimeters long and 1 centimeter square. Measure its dimensions 

carefully, and beginning at one end, 
lay off cubic centimeters after taking 
due account of the cross-sectional 
area. Drill a deep hole in the end of 
the bar and embed a nail heavy enough 
to cause the bar to float in an upright 
position, as shown in (1) Fig. 89. 
Melt paraffin into the pores of the 
wood over a flame in order to make it 
water-proof. Float the bar in water, 
and read off the number of cubic cen- 
timeters of the portion beneath the 
liquid surface, i.e. the displacement. Compare the weight of the water 
displaced with the weight of the bar itself. 

2. Counterpoise a glass beaker (or tumbler) on a balance. Prepare 
a vessel, as shown in (2) Fig. 89, to catch displaced water. Carefully 
float a piece of well-paraffined wood weighing about 100 grams and 
catch the displaced water. Wipe the block of wood dry, and place it 
on one scale pan of the balance, and pour the displaced water into the 
beaker placed on the other. The two should balance. 

If a block of wood is entirely immersed in water, the dis- 
placed water weighs more than the wood, and therefore the 
buoyant force is greater than the weight of the block. 
Hence the block will be forced toward the surface. As a 
portion of the block rises above the surface of the water, 
the displacement decreases until the weight of the block 
and that of the displaced water are equal. This fact will 
be found to be verified by either of the experiments just 
described. The law may be expressed as follows : 



Fig. 89. — Flotation Illustrated. 



MECHANICS OF LIQUIDS 



123 



A floating body sinks to such a depth in the liquid that the 
weight of the liquid displaced equals the weight of the body. 

123. Measuring Volumes by Archimedes' Principle. — 

Archimedes' principle affords an easy and accurate 
method for ascertaining the volumes 
of solid objects of irregular shapes. 
A body immersed in water evidently 
displaces a volume of water equal to 
its own volume, and, according to 
Archimedes' principle, loses in weight 
an amount equal to the weight of the 
displaced volume of water. (See Fig. 
90. ) Since 1 cubic centimeter of water 
weighs 1 gram, the body immersed 
contains as many cubic centimeters as 
the number expressing its loss of 
weight in grams. For example, a 
piece of metal that weighs 45.75 
grams in air and 40.23 grams when 
immersed in water displaces the difference, or 5.52 grams 
of water. The volume of water displaced is therefore 5.52 
cubic centimeters. Hence the volume of the immersed 
body is 5.52 cubic centimeters. When the loss of weight 
is measured in pounds, the volume expressed in cubic feet 

is found by dividing 




Fig. 90. — Weighing a 
Body in Water to Find 
its Volume. 



A 




Waterl 


— 




^ 


a 

Line 




B 




r 


H 




C...\ 












\. D. 




\ " 


1 


P 


1 


p 





that loss 
Why? 



by 62.5. 



Fig. 91. — The Floating Dry Dock. 



124. The Floating Dry 
Dock. — Among the useful 
modern inventions depend- 
ing upon the law of flotation 



(§ 122) is the floating dry dock, which is shown diagTammatically 
in Fig. 91. When the several water-tight compartments P, P, P are 
allowed to fill with water, the dock sinks until the water level is at 



124 A HIGH SCHOOL COURSE IN PHYSICS 

AB. A vessel to be repaired is then floated into the dock, and the 
water pumped out of the various compartments. As the chambers 
are emptied the dock rises sufficiently to bring the water level to 
the line CD. The boat is thus lifted clear of the water. 

EXERCISES 

1. A stone weighing 400 g. under water weighs 480 g. in air. What 
mass and volume of water does it displace? What is the volume of 
the stone? 

2. What is the volume of a metal cylinder that weighs 30 g. in air 
and 19 g. when immersed in water? 

3. A solid weighs 20 lb. in air and 12 lb. when suspended under 
water. What is the weight of an equal volume of water? What is 
the volume of the body in cubic inches? 

4. A body weighing 50 g. in air weighs 35 g. when immersed in 
water and 38 g. when immersed in oil. Find the mass and volume 
of the oil displaced. 

5. A block of iron weighing 12 g. and a piece of wood weighing 
4 g. are fastened together and weighed in water ; their weight when 
immersed is 7.5 g. If the iron alone weighs 10.2 g. when immersed 
in water, what is the volume of the wood? 

Suggestion. — From the combined volumes subtract that of the 
iron. 

6. A block of wood is floated in a vessel full of oil. If 200 g. is 
the weight of the oil displaced, what is the weight of the wood? 

7. A boat that weighs 450 lb. displaces how many cubic feet of 
water ? 

8. A ferry-boat weighing 700 tons takes on board a train weigh- 
ing 550 tons. Express the total displacement in cubic feet. 

9. Why does throwing the hands out of water cause the head of a 
swimmer to be submerged? 

10. An egg will sink in water and float in brine. A solution of 
salt may be made of such a strength that an egg will remain at any 
depth. Explain. 

11. What is the volume of a man weighing 150 lb. if he floats with 
2V of his body above water ? 

4. DENSITY OF SOLIDS AND LIQUIDS 

125. Density of a Solid. — (a) When the solid is more 
dense than water. The density of a body is defined as its 



4 



MECHANICS OF LIQUIDS 



125 



mass per unit volume and is found by dividing the number 
of units of mass by the number of units of volume (§ 12). 
In the C. G. S. system density is expressed in grams per 
cubic centimeter. For example, the density of mercury is 
13.59 grams per cubic centimeter. 

In ascertaining the density of a solid that sinks in water 
and does not dissolve, the mass is first found by weighing 
the body in air, and the volume is then measured by the 
method described in the preceding section. By definition, 

T^ . , ^ . q. mass (in grams') ^, x 

Density {in grams per cmr) = ^ r if • (i) 

volume (i7i cm.^) 

But, if the number of units of volume is found, as in 
practice, by ascertaining the apparent loss of weight sus- 
tained when the solid is submerged in water, we have for 
the numerical value of the density in the metric system, 

mass (in grams^ 



Density = 



(2) 



weight lost in water (in grams^ 

(6) When the solid is less dense than water. When a 
solid is not dense enough to sink in water, it may be 
attached to a sinker that 
is sufficiently heavy to 
submerge it. Let a 
sinker S and a light 
solid A whose mass is 
M grams be suspended 
from the arm of a bal- 
ance, as shown in Fig. 
92, so that the sinker 
alone is submerged. 
Let the weight of the 
two bodies thus arranged 
be TTj grams. Now let both solids be immersed, and the 
combined weight be W^ grams. The difference TTj — TPg 




Fig. 92. — Ascertaining the Volume of a 
Solid Less Dense than Water. 



126 A HIGH SCHOOL COURSE IN PHYSICS 

is evidently caused by the buoyant force of the water on 
the light solid A, and, according to Archimedes' principle, 
is equal to the weight of the water displaced by this 
body. This difference is numerically equal to the volume 
of the body expressed in cubic centimeters. Hence an 
equation expressing the numerical value of the density of 
the solid may be written as follows : 

Density = M (in grams-) 

Wj — W2 (m grams^ 

126. Specific Gravity. — Occasional use is made of the 
term specific gravity (abbreviated sp. gr.^ to express tlie 
heaviness or lightness of a body as compared with the 
weight of some standard substance. The specific gravity 
of any solid or liquid is the ratio of the weight of the body to 
the weight of an equal volume of pure water at 4° O. In the 
case of gases the standard with which they are compared 
is either air or hydrogen. 

Since one cubic centimeter of pure water at 4° C. weighs 
one gram, it follows that the density of a body in grams 
per cubic centimeter is numerically equal to its specific 
gravity. For example, the density of lead is 11.36 grams 
per cubic centimeter; hence lead is 11.36 times as heavy 
as an equal volume of water. Therefore, the specific 
gravity of lead is 11.36. 

Again, since the specific gravity of lead is 11.36, one 
cubic foot of lead will weigh 62.5 pounds x 11.36, or 710 
pounds. Hence in the English system of measurement 
the density of lead is 710 pounds per cubic foot. 

It is to be carefully observed that specific gravity and 
density are not the same thing ; they are numerically equal 
only when density is given in C. G. S. units. In the 
English system, density is entirely different from specific 
gravity, as the example plainly shows. 



MECHANICS OF LIQUIDS 127 

The equation of specific gravity is 

o 'J2 '4. weight of a body .^^ 

Specific gravity = — — -^ — -^ (4) 

weight of an equal vol. of water 

127. Density of Liquids. — There are several methods 
for finding the density of a liquid. Let a solid that is 
denser than water be weighed in air, then in water, and 
finally in a liquid whose density is to be found. The loss of 
weight ii^ water then represents numerically the volume 
of the submerged solid. The loss of weight in the liquid of 
unknown density represents the mass of an equal volume 
of that liquid. Therefore the density of the liquid is found 
by dividing the latter loss by the former. For example, a 
solid weighs 80 grams in air, b^) grams in water, and 60 
grams in another liquid. The mass of water displaced is 
25 grams and that of the other liquid 20 grams. Hence 
the volume of the liquid of unknown density displaced by 
the solid is 25 cubic centimeters, and its density 20 -=- 25, 
or 0.8 gram per cubic centimeter. 

128. Hydrometers. — The law of floating bodies (§ 122) 
affords a convenient method for finding the density of a 
liquid. Let the bar of wood used in Experiment 1, § 122, 
be placed in a jar of water and the volume of the sub- 
merged portion ascertained. Next place the bar in a 
liquid whose density is to be found, and again find the 
volume of the part beneath the liquid. If the displacement 
of water is Vj cubic centimeters, the weight of the displaced 
water is v^ grams. If d is the density of the other liquid 
and ^2 its displacement, the weight of the liquid displaced 
is v^d grams. According to the law of floating bodies the 
weight of the liquid displaced in each instance is equal to 
the weight of the floating bar. Hence the two displace- 
ments are of equal weight. We may therefore write 

Vi 

V2d = Vi ; whence d = — (5) 



128 



A HIGH SCHOOL COURSE IN PHYSICS 



A floating body that is made in a convenient form for 
measuring the densities of liquids is called an hydrometer. 
If the weight of the hydrometer remains con- 
stant as in the case of the wooden bar used 
in the experiment just described, the instru- 
ment is called an hydrometer of constant 
weight. In some instances the densities are 
indicated upon the bar and may be read off 
at once by observing the point to which the 
instrument sinks. Many forms of hydrom- 
eters are in daily use, the size, shape, and 
graduations being adapted to the particular 
commercial purpose that they serve. A com- 
mon form is shown in Fig. 93. A cylindri- 
cal glass tube terminates below in a small 
bulb filled with mercury, which causes the 
instrument to float in an upright position. The upper por- 
tion of the tube is made small, and the graduations are 
upon a paper scale sealed within. 




Fig. 93. — An 
Hydrometer 
of Constant 
Weight. 



Nicholson's Hydrometer. — The Nicholson hydrometer shown in 
Fig. 94 represents the type known as hydrome- 
ters of constant volume, i.e. of constant displace- 
ment. This instrument is used in finding the 
density of a solid body. Let w^ grams be the 
weights required in the upper pan A , to sink the 
hydrometer to a certain mark placed on the slen- 
der stem a. After placing the solid of unknown 
density in the upper pan, the weight required to 
sink the instrument to the same mark is W2 grams. 
Then tL\ — w^ is the mass of the solid. Let Wg be 
the weights required in the upper pan after the 
solid has been transferred to the lower pan B. 
Then w^ — w^ is the mass of the water displaced 

and numerically equal to the volume of the solid. „ IT tvt- 1 1 , 
•^ ^ Fig. 94. — Nicholson's 

Therefore, the numerical value of the density Constant Volume 
of the solid expressed in metric units is : Hydrometer. 




MECHANICS OF LIQUIDS 129 



Density = "^ ^- (6) 

Wo — Wo 



EXERCISES 

1. A piece of lead weighs 56.75 g. in air and 51.73 g. when sus- 
pended in water. Find the volume and density of the lead. 

2. A cylinder of aluminium weighs 28.35 g. in air and 17.85 g. 
when immersed in water. Calculate the volume and density of the 
metal. Compute the sp. gr. of aluminium. 

3. A piece of glass weighing 45 g. in air weighs 22.5 g. in water 
and 23.75 g. in oil. Calculate the densities of the glass and the oil. 

4. What would be the weight of the glass in Exer. 3 when im- 
mersed in a liquid whose density is 0.922 g. per cubic centimeter? 

5. The density of marble is 2.7 g. per cubic centimeter. What is 
the weight of a rectangular block 1 m. long, 40 cm. wide, and 15 cm. 
thick? Compute the sp. gr. of marble. AVhat is its mass per cu. ft.? 

6. Silver is 10.4 times as heavy as an equal volume of water. 
What will 20 g. of silver weigh when immersed in water? 

7. Ice is 0.9 as heavy as an equal volume of water. If a piece of 
ice weighing 500 g. floats on water, what is the volume of the sub- 
merged portion ? What is the volume of the ice? 

Suggestion. — Apply the law of floating bodies and ascertain the 
weight of water displaced. 

8. A bar of wood weighing 100 g. floats on water with 0.82 of its 
volume submerged; when placed in oil, 0.80 of its volume is submerged. 
Calculate the sp. gr. and density of the oil. 

9. A piece of paraffin weighs 69 g. in air and when attached to a 
sinker and suspended in water, 85.8 g. If the weight of the sinker in 
water is 95.7 g., what is the volume and density of the paraffin? 

10. The weight required to sink a Nicholson hydrometer to the 
mark on the stem is 45 g. ; when a piece of marble is placed in the 
upper pan, the weight required is 15.3 g. and with the marble in 
the submerged pan 26.3 g. Find the density of the marble. 

11. Find the density of paraffin from the following data : 
Weight required to sink Nicholson's hydrometer to mark 56.4 g. 
Weight to sink hydrometer with paraffin in upper pan 45.6 g. 
Weight required with paraffin in submerged pan 57.6 g. 

12. The density of mercury is 13.59 g. per cubic centimeter. If a 
cubic foot of water weighs 62.5 lb., what is the weight of a cubic inch 
of mercury? 

10 



130 A HIGH SCHOOL COURSE IN PHYSICS 

5. MOLECULAR FORCES IN LIQUIDS 

129. Cohesion and Adhesion. — Many of the phenomena 
of nature lead to the conclusion that bodies are made up 
of extremely minute particles to which is given the name 
molecules. The molecules of bodies possess more or less 
freedom of motion among themselves depending on the 
nature of the body ; this freedom is greatest in gases and 
least in solids. It is due to the very perfect freedom of 
motion among the molecules of a liquid (i.e. to the prop- 
erty of fluidity') that they are able to transmit pressures 
in all directions, thus giving rise to the principle stated in 
(1) § 104 and to Pascal's Law discussed in § 110. 

When near together molecules attract one another, thus 
producing the resistance found when we attempt to break 
a wire, to remove paint from glass, to tear paper, etc. 
The name cohesion is given to this attraction when it is 
between molecules of the same kind, and the term adhesion 
applies to the attraction when the molecules are of differ- 
ent kinds. 

Cohesion serves to bind individual molecules into bodies, 
or masses, and adhesion to hold together bodies of differ- 
ent kinds. Two clean surfaces of lead will cohere when 
pressed firmly together, and the dentist hammers gold leaf 
into a solid lump to form the filling for a tooth. The 
blacksmith brings together white-hot pieces of iron and 
by blows brings their molecules into such close proximity 
that cohesion takes place. Clean graphite powder is 
pressed into a solid mass to form the lead of a pencil ; the 
force of cohesion holds the particles together. The attrac- 
tion between wood and glue, stone and cement, paint and 
iron, etc., affords cases illustrating the force of adhesion. 

130. Surface Films of Liquids. — Although steel is 
nearly eight times as dense as water, a small sewing 



MECHANICS OF LIQUIDS 



131 



needle may be caused to " float " on water by placing it 
carefully upon the surface. If the surface of the liquid 
is closely examined, it will be observed that the needle 
rests in a slight depression, as shown 
in Fig. 95. In fact, the appearance 
is as if a thin membrane were stretched 
across the surface of the water. 

The three following experiments 
show an important property of the surface films 
liouids : 



Fig. 95. — An Ordinary 
Steel Needle " Floating " 
on Water. 



of 



1. Let a soap bubble be blown on the bowl of a clay pipe or the 
mouth of a small glass funnel. If the stem is left open a moment, it 
may be observed that the bubble diminishes in size. A candle flame 
held near the opening in the stem will be deflected by the current of 
air that is forced out by the contracting bubble. 

2. Let a loop of thread be tied to one side of a wire frame 4 or 5 
centimeters in diameter so that it will hang near the center, as shown 

in (1), Fig. 96. If now the frame 
is dipped into a soap solution, a 
liquid film will form across it with 
the loop closed, as shown. If, how- 
ever, the film within the loop of 
thread be broken by means of a hot 
wire, the loop instantly opens out 
into a circle, as shown in (2). 

3. Let a mixture be made of alco- 
,js y^) liol ^iid water of such strength that 

Fig. 96. — An Effect of Surface Ten- a drop of olive oil will remain in it 
sion in a Liquid Film. at any depth. The oil may be in- 

troduced beneath the surface by means of a small glass tube. It 
will be found that the globule of oil at once assumes a spherical 
form. In order to avoid an apparent flattening of the drop of oil, a 
body with flat sides should be used. 

These experiments show that the surface film of a liquid 
tends to contract and become as small as possible. In 
Experiment 1 the contraction produces a pressure that 
drives the air from the tube. In Experiment 2 the film 





132 A HIGH SCHOOL COURSE IN PHYSICS 

assumes the least possible area. This is the case when 
the area within the loop is as large as it can become, i.e. 
when it is circular. In Experiment 3 the film of oil com- 
pletely incloses the globule and causes the oil to assume 
the geometrical form requiring the least superficial area 
for a given volume. The spherical form fulfills this con- 
dition. For a similar reason a soap bubble takes the 
shape of a sphere. 

131. Surface Tension. — It will be seen from a study of 
Fig. 97 that the conditions under which the surface mole- 
cules of a liquid exist is very 
different from that of the mole- 
cules lower down. Molecule 
A at the center of the small 
circle is attracted equally in all 

Fig. 97. — Molecular Forces near directions by the neighboring 
the Surface of a Liquid. molecules. This is the force of 

cohesion and acts across a very small distance represented 
here by the radius of the circle. Very near the surface the 
forces acting downward on molecule B are greater than 
those that act upward, while for molecule C there is no 
upward attraction whatever. Hence the surface layers 
of molecules are greatly condensed by this excess of force 
acting always toward the body of the liquid. In this man- 
ner is formed a tough, tense surface film whose constitution 
is different from that in the interior mass. The measure 
of the tendency of the surface layers of a liquid to contract 
is called surface tension. 

132. Capillary Phenomena. — The effects of surface 
films were first investigated in small glass tubes of hair- 
like dimensions. Hence the name " capillarity " arises 
from the Latin word eapillus, meaning "hair." Figure 
98 represents a series of glass tubes varying from 0.2 mil- 
limeter to 2 millimeters in diameter. If the tubes are 



MECHANICS OF LIQUIDS 



133 




in Capillary 
Tubes. 



moistened and then set upright in a shallow vessel of 
water, the liquid will rise in the tubes, — highest in the 
smallest tube and least in the largest. It 
may also be observed that the surface of the 
liquid turns upward wherever it comes in 
contact with the glass. Hence the surface 
within the tube is concave, and a so-called 
meniscus is formed. 

If, now, glass tubes of small bore are 
placed in mercury (Fig. 99), the liquid, 

which in this case does ^^^;. ^^- "~?3®^^: 

tion of a Liquid 

not wet the glass, will 
be depressed within 
the tubes. The depression is great- 
est in the smallest tube and least in 
the largest. The edges of the film 
in contact with the glass may be 
seen to turn downward so that the 
surfaces within the tubes are convex, 
and the meniscus thus formed is in- 
verted. 
The following laws of capillary action may be stated : 

(1) Liquids are elevated in tubes which they wet^ hut are 
depressed in tubes which they do not wet. 

(2) The elevation or depression is inversely proportional 
to the diameter of the 

VA 

UZ) 




Fig. 99. — Depiession of 
Mercury in Capillary 
Tubes. 



tubes, 

133. Capillary Ac- 
tion Explained. — If 

a plate of glass, shown 
in cross section in 
(1), Fig. 100, be 
placed upright in a 



%. 



m< 



# 

(i) (2) 

Fig. 100. — Water Lifted by the Contraction of 
the Surface Film. 



m 



134 A HIGH SCHOOL COURSE IN PHYSICS 

shallow vessel of water after having been moistened, 
there will be formed a continuous film ABO lying partly 
upon the glass and partly upon the water. On account 
of the tendency of this film to contract, as shown in § 131, 
the corner at B will be rounded and a small portion of the 
water lifted against the glass. 

Again, if a moistened tube of glass, (2), Fig. 100, be 
dipped into water, a film AB CD is formed adhering to the 
glass and extending across the water in the tube. Ow- 
ing to the tendency of this film to contract, a force is pro- 
duced that is sufficient to lift the column of water BEFQ 
in opposition to the force of gravity. When the tube is 
dipped into mercury, the liquid forms no film adhering to 
the glass above the surface level. On the other hand, the 
surface film of mercury (see Fig. 99) continues down- 
ward along the glass walls of the tube and then upward 
into the tube at the lower end. The force which is 
developed by the contraction of this film tends to pull 
the liquid down within the tube. The depression thus 
produced is such that the downward force of the con- 
tracting film is balanced by the upward pressure of the 
liquid within the tube. 

134. Capillary Action in Soils. — The principles of 
capillary action find an important application in the 
distribution of moisture in the soil. In compact soil 
water is brought to the surface in much the same manner 
as it rises in a piece of loaf sugar which is allowed to 
come in contact with it. As rapidly as evaporation goes 
on at the surface, the loss is supplied by capillarity from 
below. In dry weather it is desirable to prevent this 
surface loss, which is done by " mulching " and loosen- 
ing the soil by cultivation. In this latter process the 
grains of soil are broken apart and the interstices thus 
made too large for effective capillary action to take place. 



MECHANICS OF LIQUIDS 135 

The moisture then rises to a level a few inches beneath 
the surface, where it is made use of by growing plants. 

EXERCISES 

1. Why is a drop of dew spherical ? Examine a small globule of 
mercury placed on glass. How do you account for its form ? 

2. By heating a piece of glass until it softens the sharp corners 
become rounded and smooth. Explain. 

3. In the manufacture of shot molten lead is poured through 
a small orifice at the top of a tower. In falling the stream breaks 
up into drops which solidify before reaching the earth. What gives 
the spherical form to these masses of lead ? 

4. Why does oil flow upward in the wick of a lamp? Will 
mercury do the same? 

5. Grease may be removed from a piece of cloth by covering it 
with blotting paper and passing a hot flatiron over it. Explain. 

6. Place two toothpicks upon water about a centimeter apart. 
Touch the liquid surface between them with a glass rod moistened 
with alcohol. From the manner in which the pieces of wood move 
about, observe which liquid film has the greater tension. 

7. Explain how an insect can run on the surface of water without 
sinking. 

8. Explain the action of a towel ; of blotting paper ; of sponges. 

9. Why can we not write with ink upon unglazed paper? 

10. Why are the footprints made in newly cultivated soil moist 
while the loose earth is dry ? 

11. Explain why it is necessary to pack the earth around plants 
and small trees when they are first set out. Later on we loosen the 
soil about them. Why ? 

SUMMARY 

1. The force exerted by a liquid of uniform densit}^ 
against a given surface is directly proportional to the 
depth of the surface (§ 103). 

2. At a given depth the force exerted by a liquid is the 
same in all directions and acts always in a direction per- 
pendicular to the surface against which it presses (§ 104). 

3. The force exerted by a liquid on any horizontal sur- 
face is equal to the product of the area and depth of the 



136 A HIGH SCHOOL COURSE IN PHYSICS 

surface and the density of the liquid. The equation is 
total pressure = ahd (§ 106). 

4. When the surface pressed against is not horizontal, 
the product of the area and the density must be multiplied 
by the depth of the center of mass of the given sur- 
face (§ 106). 

5. The force exerted by a liquid against the bottom of 
a vessel of given depth is independent of the form of the 
vessel (§ 107). 

6. A liquid at rest in communicating vessels remains 
at the same level in all its parts (§ 109). 

I 7. A force applied to any area of a confined liquid is 

\^y^' I transmitted undiminished by the liquid to every equal 

^ I area of the interior of the containing vessel and to every 

1 part of the vessel. This is known as Pascal's Law (§ 110). 

8. The mechanical advantage of the hydraulic press is 
the ratio of the area of the large piston to that of the small 
one (§ 111). 

9. A body immersed in a liquid is buoyed up by a 
force equal to the weight of the liquid that it displaces. 
This is known as Archimedes' Principle (§ 120). 

10. A floating body displaces a mass of the liquid in 
which it floats, whose weight equals its own (§ 122). 

11. The volume of a body insoluble in water may be 
found, according to Archimedes' Principle^ by ascertaining 
the weight of water that it will displace (§ 123). 

12. Density = -^^^^^ (§ 125). 

volume 

13. Bodies are assumed to be composed of extremely 
small particles called molecules. When near together, 
molecules attract each other. Cohesion is the attraction 
between molecules of the same kind ; adhesion is the at- 
traction between molecules of different kinds (§ 129). 



MECHANICS OF LIQUIDS 137 

14. The surface of a liquid tends to contract and be- 
come as small as possible; hence the spherical form as- 
sumed by soap bubbles, dew drops, globules of mercury, 
etc (§130). 

15. Liquids are elevated in tubes which they wet, but 
are depressed in those which they do not wet. The ele- 
vation or the depression is inversely proportional to the 
diameter of the tube (§ 132). 

16. Moisture rises readily in compact soils, but ceases 
to rise in those of loose texture. Hence the loss of soil 
water by evaporation is effectually prevented by " mulch- 
ing" or by cultivation (§ 134). 



CHAPTER VIII 
MECHANICS OF GASES 

1. PROPERTIES OF GASES 

135. Characteristics of Gases. — Gases, like liquids, 
possess the iproipeTtj oi fluidity, i.e. they may be deformed 
by any force however small. But a gas differs from a 
liquid in that it has no definite size of its own ; it not 
only fits itself to the shape of the vessel containing it, 
but always entirely fills it. On account of their common 
property of fluidity, liquids and gases are classed together 
as fluids. As a consequence of this property, the laws of 
pressure relative to liquids stated in 1, § 104 and in § 110 
are equally applicable to gases. Other laws, however, 
arise in the case of a gas on account of the tendency to 
adapt its size to the capacity of the containing vessel. 

136. Laws Common to Liquids and Gases. — (1) The 
force exerted against any surface is perpendicular to 
that surface. (2) The force at any point is the same in 
all directions (§ 104). (3) An immersed body is buoyed 
up by a force equal to the weight of the liquid or the gas 
that it displaces (§ 120). 

137. Weight and Density of Air. — We are taught by 
everyday experience that a gaseous medium which we 
call air surrounds us on every hand. The bubbles that 
may be produced by blowing through a tube inserted into 
water, the process of breathing, the resistance that the air 
offers to a rapidly moving bicycle or train, the various 
effects of the wind, and many other phenomena demon- 
strate to us the presence of this medium, 

138 



MECHANICS OF GASES 139 

It was shown by experiment in § 3 that air has weight. 
The same apparatus may be used in finding the density of 
air in the following manner: 

Let the bulb be weighed carefully before and after admitting the 
air. The increase in weight will give the mass of the air contained in 
the bulb. The capacity of the bulb is next ascertained by filling it 
with water and weighing it. Dividing the mass of the air by the 
capacity of the bulb gives the density of the air. 

The density of air at the temperature of freezing water 
and under the average sea-level pressure is 0.001293 gram 
per cubic centimeter. Hence a liter (1000 cm.^) of air 
weighs nearly 1.3 grams. A cubic foot of air weighs about 
an ounce and a quarter ; hence 12 cubic feet weigh nearly 
a pound. The amount of air in an ordinary schoolroom 
weighs more than half a ton. Thus we live submerged 
in an ocean of air which extends many miles above us and 
exerts a pressure of nearly 15 pounds per square inch. 

2. PRESSURE OF THE AIR AGAINST SURFACES 

138. Atmospheric Pressure. — Since air has weight and 
fluidity, the atmosphere must exert a force against all 
surfaces with which it comes in contact. The existence of 
such a force may be shown by the following experiments : 

1. Fill a tumbler with water and invert it in a vessel of water. 
Lift the inverted tumbler until its opening is horizontal and just 
submerged, as in Fig. 101. The water re- 
mains in the tumbler because it is sup- 
ported by the force exerted by the atmos- 
phere downward against the free surface 
of water in the vessel. 

2. While the inverted tumbler is in the 

condition shown in Fig. 101, place a piece 

of cardboard across its mouth, pressing it 

close against the rim. Carefully lift the 

4. 1 1 4- ,1 , 1,1 J, 1 Fig. 101. — Atmospheric 

tumbler from the water, and the cardboard tj c- \.- 

rressure Supporting 

will not fall off. The force exerted by Water. 




140 



A HIGH SCHOOL COURSE IN PHYSICS 





the air against the cardboard in an upward direction is sufficient to 
support the weight of the water in the tumbler, as in Fig. 102. 

3. Tie a piece of sheet rubber over a 
"^ glass vessel, as shown in Fig. 103. Place 

»^ ^, the vessel on an air pump and exhaust 

'f/^ ,. /NA^ th® ^ii' from the 

space beneath 

-^m-Mff *^^^ rubber. The 

''''"" membrane, no 

longer supported \ w, . ,.., ,,, 

^ .. • f ^ — ^X--\ i^ 

by the air from '^ ^^^^ ~ 

below, is more 
Fig. 102. — Action of Atmos- , , 

pheric Pressure Upward. ^"^ ^^°^® ^^" 

pressed until it 

finally bursts under the pressure of the air 

, Fig. 103. — Atmospheric 

above. t> t> i u 

Pressure upon a Rubber 

4. If a glass tube 3 or 4 feet long (or Membrane, 

even longer) can be secured, place it nearly 

vertical with one end in a tumbler of water. Try to elevate the 
water to the top of the tube by " sucking " on the upper end of 
it. Now insert the lower end of the tube in mercury and try to 
elevate it in the same manner. It is so much denser than water, 
that it can be lifted only a few inches. 

139. Limitations of Atmospheric Pressure. — The atten- 
tion of Galileo was called to the fact that in wells of 
unusual depth suction pumps (§ 155) were unable to lift 
water more than about 32 feet above the level of the water 
in the wells. At that time it was supposed that " nature 
abhorred a vacuum " and that she hastened to fill all such 
spaces with whatever material happened to be most avail- 
able. Thus Galileo was led to believe that nature's abhor- 
rence for a vacuum had its limitations, and he probably 
suspected that the rise of water in pumps was due to air 
pressure on the surface of the water. It remained for his 
pupil, Torricelli, however, to devise a suitable method for 
measuring the actual pressure of the atmosphere. He suc- 
ceeded in doing this in the year 1643. 



MECHANICS OF GASES 



141 




140. Torricelli's Experiment. — This important histori- 
cal experiment is performed by making use of a strong 
glass tube about 80 centimeters long which is closed at one 
end. The tube is first filled with 
mercury in order to expel the air, 
after which it is closed by the 
finger and inverted in a vessel of 
mercury, Fig. 104, care being 
taken to prevent the entrance of 
air. The mercury falls at once 
and leaves a vacuum of several 
centimeters at the top of the 
tube. The force due to the at- 
mosphere which is exerted down- 
ward against the free surface of 
mercury in the vessel is trans- 
mitted to the interior of the tube ^^^- ^^^ 
in which it is able to support 
a column of the liquid usually about 75 centimeters in 
height. 

141. Pascal's Experiment. — Pascal reasoned that if the 
column of mercury in a Torricellian tube were indeed 
supported by the atmosphere, the column should become 
shorter at a high altitude. The column was accordingly 
measured at the top of a high tower in Paris and a small 
decrease detected. Five years after Torricelli's discovery, 
the tube was carried to the top of the Puy de Dome, a 
high mountain in Auvergne, France, where a test showed 
a decided decrease of about 8 centimeters in the height of 
the mercurial column. 

142. Variations in Atmospheric Pressure. — The atmos- 
pheric pressure at a given place is far from being constant. 
The variations at any given place amount to about 3 cen- 
timeters. At sea-level the average height of the mercurial 



Torricelli's Experi- 
ment. 



142 



A HIGH SCHOOL COURSE IN PHYSICS 



column is about 76 centimeters ; hence this height is taken 
as the standard of pressure and is called mean sea-level 
pressure. Any pressure equivalent to a column of mer- 
cury 76 centimeters high is said to be "one atmosphere." 
143. The Barometer. — The mercurial barometer (pro- 
nounced harbm'eter) is merely a mounted Torricellian 
tube for showing the pressure of the atmos- 
phere. In order that it may give correct 
indications, the mercury must be pure and 
clean, and the space above the liquid in the 
tube as free as possible from the presence of 
air and other gases; i.e. the vac- 
uum must be as nearly perfect 
as possible. Atmospheric pres- 
sure is usually measured in inches 
or centimeters of ynereury. The 
space above the mercury in the 
tube is known as the Torricellian 
vacuum. 

As the height of the mercurial 
column changes with the pressure 
of the air, the surface of the 
liquid rises and falls in the* res- 
ervoir, or cistern, below. For an 
accurate reading of the barometer 
the mercury column must be meas- 
ured from the surface of the liquid 
in the reservoir. In the barome- 
ter shown in Fig. 105 the reser- 
voir has a flexible bottom. By 
turning the screw /S' the surface 
of the mercury in the cistern is 
brought to the zero point of the scale, whose position is 
marked by an ivory index B. The height of the mercury 




Fig 



105. — A Standard Mer- 
curial Barometer. 



MECHANICS OF GASES 



143 



/?^^ 



& 



® 



is then read by observing the position of the 
upper surface of the liquid in the tube at A. 

Some mercurial barometers have the form 
shown in Fig. 106. The height of the mer- 
cury is found by reading the positions of the 
two liquid surfaces A and B^ in which case 
the difference between the two readings gives 
the length of the mercury column supported 
by the atmosphere. 

144. The Aneroid Barometer. — A barometer of 

the form shown in Fig. 107 is in common use. As the 
name indicates, the aneroid barometer is " without 
liquid," and depends for its operation on a circular 
chamber C 
made of thin 
metal with 
corrugated 
sides. The 
air in the 
chamber is 
partly re- 
moved, after 
which the chamber is hermetically sealed. An increase 
of atmospheric pressure forces the side of the box 
inward, but a decrease allows it to spring out. 
This motion, 




(2) 

The Aneroid Barometer. 



Fig. 106. — a 
Mercurial 
Barometer. 



although very 
slight, is 
transmitted 
through the 
multiplying systems of levers 
L aud A and the small chain 
B to the axle S which car- 
ries the index, or movable 
pointer, /. The scale is 
graduated to correspond to 
the readings of a standard 
mercurial barometer. These 




Fig. 108. — The Barograph, or Self-registering 
Barometer. 



144 



A HIGH SCHOOL COURSE IN PHYSICS 



instruments are made of convenient size to be used by surveyors and 
explorers in ascertaining altitudes. It should be said that the words 
*' Rain," " Fair," etc., printed on the dial of an aneroid barometer are 
merely indications of the general trend of weather changes. 

The principle of the aneroid is employed in the " barograph," or 
self-recording barometer, shown in Fig. 108. This instrument is 
provided with an index which carries a pen that makes a continuous 
record of the atmospheric pressure on a revolving cylinder covered with 
suitable paper. A portion of such a record is shown in Fig. 109. 




Fig. 109. — Portion of a Record Made by a Barograph. 



145. Utility of the Barometer. — The barometer is an 
important laboratory instrument, inasmuch as the atmos- 
pheric pressure must be known in carrying out many 
experiments in both Physics and Chemistry. In this 
respect it ranks in usefulness with the thermometer, 
balance, etc. 

From the readings of barometers taken simultaneously 
at many places of observation and telegraphed to central 
stations, the direction of atmospheric movements can be 
predicted. Thus the barometer becomes an aid in fore- 
casting the weather. Furthermore, a "low" barometer, 
i.e. decreased pressure, usually accompanies or precedes 



MECHANICS OF GASES 



145 



stormy weather, while a rising barometer generally de- 
notes the approach of fair weather. If a weather map 
be consulted, certain regions will be found marked 
"High" and others marked "Low." The places at which 
the pressures are equal are joined by curves called 
isobars^ upon each of which is indicated the barometric 
reading. Fig. 110. 
The direction of the 
wind at each place of 
observation is indi- 
cated by an arrow. 
The general direc- 
tion of the wind is al- 
ways from places of 
"high" toward those 
of " low " pressure. 

Another important 
use of the barometer 
is made in measuring 
the difference in alti- 
tude of two places. 
This measurement de- 
pends upon the fact 
that the atmospheric 
pressure decreases with the elevation above sea-level. 
For places not far above the level of the sea the decrease 
is about 1 millimeter for every 10.8 meters of elevation, 
or 0.1 inch for every 90 feet of ascent. The decrease in 
pressure as one climbs a mountain is easily accounted for 
when it is recalled that it is the air above the level of 
a given place that produces the pressure. Descending 
a mountain or into a mine simply submerges one more 
deeply in the atmospheric ocean and thus puts above his 
level more air to be supported. 
11 



^1 29.6 \ 


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29W^S^^ 


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^^^ 


7?^ 

s 


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m^ 


Xlanft /7s 


^w 


^ 1 \_H\ \^'j 




^^^[^^^^° 


-^^--iV'^xHwt 


^^'^ 


^^y 29. \ 


K^VbiA Vwl 


pnplTP 


^^w\ 


\r\ /v^V^*\ 


if 


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I I 


^"^Jt? I 


Y/nA'^''^m\ \ W 


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/-A 


ig^LA 


kTl 


1\ yjT 


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. 1— 


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— 29.2 \ 




V$$it.29.4 \ 



Fig. 110. — A Portion of a Weather Map. 



146 A HIGH SCHOOL COURSE IN PHYSICS 

146. Atmospheric Pressure Computed. — In order to 
compute the pressure of the atmosphere in grams per 
square centimeter, we have only to find the pressure per 
unit area due to mercury when its depth is equal to the 
height of the barometric column. Consider a tube whose 
cross-sectional area is 1 square centimeter and in which 
the height of mercury is 76 centimeters. The pressure at 
the bottom of such a tube is the product of the area, height, 
and density of the mercury, as shown in § 106. We have, 
therefore, 1 x 76 x 13.6, or 1033.6 grams. Hence, if the 
barometer reading is 76 centimeters, the atmospheric pres- 
sure is 1033.6 grams per square centimeter. 

EXERCISES 

1. The column of mercury in a barometer stands at a height of 
74.5 cm. What is the height in inches? 

2. How high a column of water could be supported by atmos- 
pheric pressure when the barometer reads 75 cm.? 

3. When the barometer reads 74 cm., what is the atmospheric 
pressure expressed in grams per square centimeter? in dynes per 
square centimeter? 

4. If the pressure of the air is 15 lb. per square inch, calculate 
the total force exerted upon a person the area of whose body surface 
is 16 sq. ft. 

5. A soap bubble has a diameter of 4 in. Calculate the force 
exerted by the air against its entire surface when the barometer reads 

29 in. A cubic inch of mercury weighs 0.49 lb. 

6. What result would be obtained by performing 
TorriceUi's experiment with a tube twice as long as the 
one described in § 140? 

7. How would the height of mercury be changed 
if a tube of larger cross-sectional area were used ? 

8. Show why a change in the area of the mercurial 
surface in the cistern of a barometer has no effect on the 
height of the column. 

9. What result would Torricelli have obtained in his 
experiment if he had used a tube only 70 cm. long? 

Fig. in. 10- Try to suck the water from a bottle (see Fig. Ill) 




MECHANICS OF GASES 



147 




Fig. 112. — Pneu- 
matic Inkstand. 



out of which a glass tube passes through a tightly fitting rubber 
stopper. Explain the results observed. 

11. Inkstands are sometimes made in the form 
shown in Fig. 112. Explain why the ink in the 
reservoir can remain ^t a greater height than that 
outside. 

12. Why is it necessary to make a small vent 
hole in the upper part of a cask when it is 
desired to draw off the liquid in a steady 
stream from a faucet placed near the bottom ? 

13. Explain why water will not run in 
a steady stream from an inverted bottle. 

14. The Magdeburg hemispheres shown 
in Fig. 113 are closely fitting hollow vessels 
about 4 in. in diameter. By placing the 
hemispheres together and exhausting the 
air the pull of two strong boys is scarcely 
sufficient to separate them. Explain. 

15. Assuming the atmospheric pressure 
to be 15 lb. per square inch, calculate the 
force required to separate the hemispheres 

113 _ Maedeburff s^iown in Fig. 113 when the air within them 
Hemispheres. is completely exhausted. 




Fig 



3. EXPANSIBILITY AND COMPRESSIBILITY OF GASES 



147. Compressibility of Air and Other Gases. — Gases, 
unlike liquids, are easily reduced in volume by increasing 
the pressure under which they exist. This is evident 
from the fact that the quantity of air in the pneumatic tire 
of a bicycle, for example, may be increased to double or 
triple the original mass. Again, the air in a pneumatic 
cushion is compressed into a smaller space when one sits 
upon it, but it springs back to its original volume when 
the pressure is relieved. Thus air and other gases mani- 
fest the property of expansibility as well as eompressihility , 
The popgun and air rifle make use of these properties of 
air ; first the air is compressed in the cylinder of the gun. 



148 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 114. — Il- 
lustrating 
Expansi- 
bility of a 
Gas. 




then as the pellet moves, the force of expansion drives the 
missile with great acceleration from the barrel. 

1. Place a partially inflated balloon under the receiver of an air 
pump. As soon as the pump is set in action, the swelling of the 
balloon will indicate the expansive tendency of the aii* 
within it. 

2. Arrange a bottle as shown in Fig. 114. Blow 
forcibly into the tube, thus causing some bubbles of 
air to pass into the bottle above the liquid. Quickly 
remove the lips from the tube, 
and water will be driven out by 
the expanding air within the 
bottle. 

3. Place two bottles under the 
receiver of an air pump as shown 
in A and 5, Fig. 115. A is tightly 
corked and about two thirds full 
of water. B is uncorked and con- 
tains air only. When the pump Fig. 
is put into operation, the air pres- 
sure on the surface of the water in B is reduced, 
and the consequent expansion of the air in A above the liquid forces 
the water through the tube into B. It air be now admitted into the 
receiver, the water will be driven back into A . Why ? 

The expansibility of gases is explained by assuming that 
their molecules (§ 129) are in rapid motion. As a con- 
sequence of this motion, innumerable molecules strike 
against every part of the walls of the containing vessel. 
Although the mass of a molecule is extremely small, its 
speed is so enormous (equaling or exceeding that of a 
cannon ball) that it strikes the side of the vessel with an 
appreciable force. Thus a continuous storm of such blows 
results in the production of a steady outward pressure 
against the walls of the vessel inclosing the gas. 

148. Pressure and Elastic Force in Equilibrium. — The 
experiments just described teach us why hollow bodies, 



115. — Air in 
A Forces Water 
into B. 



MECHANICS OF GASES 



149 



such as balloons, cardboard boxes, etc., are not crushed by 
atmospheric pressure. The crushing force exerted against 
the external surface of a balloon only 10 centimeters in 
diameter is more than 300 kilograms. This force, how- 
ever, is counteracted by the expansive force of the gas 
within it. If the external force is decreased, the gas ex- 
pands until the forces are again in equilibrium. On the 
other hand, if the external "force is increased, the gas is 
compressed until the inward and outward forces balance 
each other. Thus the human body is able to withstand 
the enormous force exerted by the atmosphere upon it. 
All the cavities that might otherwise collapse are pre- 
vented from doing so by the expansive force of the gases 
which they contain. 

149. Law of Expansion and Compression. — The rela- 
tion which exists between the pressure to which a gas is 
subjected and its volume was 
discovered by Robert Boyle 
(1627-1691) of England, in 
1662, and by Mariotte, of 
France, fourteen years later. 
In France this relation is usu- 
ally called " Mariotte's Law," 
but we speak of it as Boyle^s 
Law. An experiment to illus- 
trate Boyle's method of dis- 
covery may be performed as 
follows : 



Take a bent glass tube (1), Fig. 
116, of which the shorter arm is 
hermetically sealed. The long arm, 
left open at the top, should be about 
90 centimeters long. The length ^iq 
of the short arm should be at least 




116. — Ilhistrating 
Law. 



Boyle's 



150 A HIGH SCHOOL COURSE IN PHYSICS 

10 centimeters. Pour a small quantity of mercury into the tube so 
that the liquid stands at the same level in both sides. We thus 
have a quantity of air confined in the space ylC under one atmosphere 
of pressure owing to the transmission of pressure by the mercury. 
Measure the length of the space AC, and then pour mercury into 
the tube (see (2), Fig. 116), until the surface of tlie liquid in 
the long arm is as far above that in the short arm as the height of 
mercury in the barometer. If the new volume of air in the short 
arm be measured, it will be found to be just one half the original 
volume. The new pressure is due both to the atmospheric pressure 
on the mercury at B and the column of mercury in the tube. It is, 
therefore, two atmospheres. 

This experiment teaches us tliat by increasing the pres- 
sure upon a confined gas from one atmosphere to two 
atmospheres, the volume is caused to be only one half as 
great. This is only a special instance, however, of a more 
general law. If p is the pressure of a gas whose volume 
is V, then 

under a pressure 2p the volume of the gas will he^ v ; 
under a pressure 3p the volume of the gas will be J v ; 
under a pressure ^p the volume of the gas will be 2 v ; 
under a pressure J p the volume of the gas will be 3 v, etc. 

Now, since 
pv = 2p X lv = Sp X ^v = J^ X 2v = ^p X 3v, 
we are able to state the relation as follows : 

The product of the pressure and volume of a given mass of 
gas at a constant temperature is constant. 

Where P is the pressure of the gas when its volume is 
V, and p the pressure when the volume is v, the law may 
be expressed algebraically in the forms : 

PV = pv, or P : p : : V : V. (l) 

Example. — The observed volume of a gas is 30 cm.^ when the 
barometer reads 74.5 cm. What volume will it occupy under a baro- 
metric pressure of 76 cm. ? 



MECHANICS OF GASES 



151 



Solution. — According to Boyle's Law the product of the pres- 
sure and volume under the first condition will be equal to their prod- 
uct under the second condition. Hence, if x represents the volume 

required, we have 

74.5 X 30 = 76 X X. 

Solving this equation for x, we obtain 

X = 29.4 cm.3. 

EXERCISES 

1. If the volume of a certain gas is 200 cm.^ when its pressure is 
1000 g. per square centimeter, what volume will it occupy when its 
pressure has been increased to 1200 g. per square centimeter ? 

2. The volume of an air bubble at a depth of 1 m. of mercury is 
1 cm.^ What will be its volume when it reaches the surface if the 
barometer reading is 75 cm. ? 

Suggestion. — The first pressure is equal to the sum of the atmos- 
pheric pressure and that due to the mercury in which it is immersed, 
or 175 cm. 

3. A gas is often confined in a tube, as shown 
in A, Fig. 117, whose open end is beneath the sur- 
face of some liquid. How much is l^he pressure of 
the gas confined in such a tube. in a vessel of mer- 
cury when the surface in the tube is 25 cm. below 
the level of the liquid outside, the barometer read- 
ing 75 cm.? 

4. If the volume of the gas under the conditions 
given in Exer. 3 is 15 cm., what will be its volume 
if the tube is elevated until the surfaces are at the 
same level ? 

5. In B, Fig. 117, the surface of the mercury in 
the tube is 25 cm. above that of the mercury on 
the outside. If the atmospheric pressure is 75 cm., 
what is the pressure of the confined gas? 

6. Under the conditions given in Exer. 5 the volume of the gas 
confined in the tube is 50 cm.^. What volume will the gas occupy 
when the surfaces are brought to the same level? 

7. The volume of an air bubble 136 cm. under water is 0.5 cm.^. 
Barometer reading = 75.4 cm. Calculate (1) the pressure to which 
the bubble is subjected, and (2) the volume it will have as it emerges 
from the water. 




Fig. 117. 



152 A HIGH SCHOOL COURSE IN PHYSICS 

Suggestion. — Reduce the depth of water to its equivalent in 
terms of mercury, and compute the first pressure as in Exer. 2. 

8. To what depth would the inverted tumbler shown in Fig. 1 
have to be taken in order to become half filled with water if the 
barometer reading is 75 cm. ? 

9. A gas tank whose capacity is 3.5 cu. ft. is filled with illumi- 
nating gas until the pressure is 225 lb. per square inch. How many 
cubic feet of gas at atmospheric pressure will be required in the filling 
of the tank? (Assume one atmosphere to be 15 lb. per square inch.) 

10. Why is it safer to test an engine boiler by pumping in water 
rather than air? 

150. Changes in Density. — Since a change in the volume 
of a given mass of gas occurs whenever the pressure is 
changed, it follows that there will also be a change in 
its density. Forcing the gas into one half its original 
volume doubles the amount in each cubic centimeter ; 
making the volume one third multiplies the mass in each 
cubic centimeter by three ; and so on. Therefore, doub- 
ling the pressure of a gas, thus reducing the volume one 
half, doubles the density ; tripling the pressure multiplies 
the density by three ; and so on. Hence, we may state 
these relations as follows : 

The density of a gas at constant temperature is directly 
proportional to the pressure. 

Expressed algebraically, 

D : d : : P : p, (2) 

where D is the density of the gas when the pressure is P„ 
and d the density when the pressure is p. 

EXERCISES 

1. Hydrogen, whose density is 0.09 g. per liter under one atmos- 
phere of pressure, is condensed in a steel cylinder until the pressure 
is 15 atmospheres. Calculate the density of the gas in the cylinder. 



MECHANICS OF GASES 153 

2. Illuminating gas is condensed in a reservoir until its density 
has increased from 0.75 g. per liter to 4.5 g. per liter. Calculate the 
pressure in the reservoir. Express the result in atmospheres. 

3. If 4 liters of air at ordinary atmospheric pressure are admitted 
into a vacuum of 10 liters capacity, what will be the pressure and 
density of the air? (Under one atmosphere the density of air is 
1.29 g. per liter.) 

4. What is the weight of the quantity of illuminating gas con- 
densed in a cylindrical tank of 3 cu. ft. capacity until the pressure is 
225 lb. per square inch ? (The density of the gas under one atmos- 
phere of pressure is 0.75 g. per liter.) 

5. If the gas shown in the tubes in Fig. 117, A and B, is air, what 
is the density of it under the conditions given in Exercises 3 and 5 
on page 151 ? 

4. ATMOSPHERIC DENSITY AND BUOYANCY 

151. Atmospheric Density Changes with Altitude. — The 
air, unlike the water of the ocean, which is practically 
incompressible, diminishes in density as one ascends a 
mountain or rises in a balloon. As the pressure becomes 
less, the density of the air decreases proportionally. Thus 
at the summit of Mont Blanc in Switzerland, an altitude 
of three miles, the barometer indicates only one half as 
much atmospheric pressure as at the sea-level. Hence the 
density of the air at this altitude is only one half as 
great. 

Aeronauts have succeeded in ascending to an altitude 
of about 7 miles, where the pressure is only 18 centimeters, 
or about a quarter of sea-level pressure. Greater altitudes 
have been explored by the aid of balloons equipped with 
self-registering instruments until a height of about 14 
miles has been attained. Figure 118 shows the changes in 
the pressure and density of the atmosphere at various alti- 
tudes. The numbers at the left indicate altitudes in miles 
above sea-level, those at the extreme right the densities of 
the air compared with the density at sea-level, while the 



154 



A HIGH SCHOOL COURSE IN PHYSICS 



next column gives tlie barometric readings in inches. From 
the figure it may be seen that at an elevation of 15 miles, 
for example, the density of the air is only one thirtieth of 




1000 30000 



3d 



Fig. 118. — Showing the Decrease of Atmospheric Pressure with Altitude. 

its sea-level density, while the barometer would read about 
one inch of mercury. 

152. Buoyancy of Air. — On account of the fact that 
the pressure of the air decreases as the altitude increases, 
its pressure downward upon the top surface of a box, for 
example, is less than its upward pressure against the bot- 
tom. It follows, therefore, as for bodies immersed in 
water (§ 121), that an object is buoyed up by a force equal 
to the weight of the air that it displaces. In general, 
bodies are so heavy in comparison with the amount of air 
displaced that the consequent loss of weight is not taken 
into account. A man of average size, for instance, is 
buoyed up by a force equal to about 4 ounces. If, how- 



MECHANICS OF GASES 155 

ever, the air displaced by a body should weigh more than 
the body itself, it would be lifted by the air just as a piece 
of wood is lifted when immersed in water. This is the 
case of a balloon, in which the weight of the material com- 
posing the bag, together with the gas used to fill it, ropes, 
basket, ballast, etc., is less than that of the air which it 
displaces. In order that this may be so, it is necessary 
to inflate the balloon with a gas lighter than air, which 
weighs 1.29 kilograms per cubic meter. Hydrogen gas is 
sometimes used whose density is 0.09 kilogram per cubic 
meter, but more frequently the material is common illumi- 
nating gas weighing about 0.75 kilogram per cubic meter. 
The balloon United States which won the first international 
race at Paris in 1906 was filled with over 2000 cubic meters 
of illuminating gas, thus creating a lifting force of more 
than a ton. 

EXERCISES 

1. A balloon whose capacity is 1000 m.^ is filled with hydrogen. 
If the weight of the bag, basket, and ropes is 235 kg., what additional 
weight can the balloon lift ? 

Suggestion. — Find the difference between the weight of the air 
displaced by the balloon and the entire weight of the balloon and its 
equipment. 

2. What will be the lifting capacity of the balloon in Exer. 1 when 
filled with illuminating gas? 

3. When will a balloon cease to rise ? When will it begin to fall ? 

4. A kilogram weight of brass (density 8.3 g. per cm.^) will weigh 
how much in a vacuum ? 

Suggestion. — Add to the weight of the body the weight of the air 
that it displaces. 

5. APPLICATIONS OF AIR PRESSURE 

153. The Air Pump. — The air pump is used to remove 
the air or other gases from a closed vessel called a receiver. 
It was invented about 16i^0 by Otto von Guericke (1602- 



156 



A HIGH SCHOOL COURSE IN PHYSICS 



i 






1686), burgomaster of Magdeburg, Germany. A simple 
form of the air pump is shown in Fig. 119. (7 is a cylin- 
der within which slides the tightly fitting piston P, R is 
the receiver from which the air is to be exhausted. The 
receiver is connected with the cylinder by the tube T. 
s and t are valves opening upward. The operation of 
the pump is as follows : 

When the piston is pushed down, the valve s permits 
the air in the cylinder to escape, but closes to prevent its 

return when the piston 
is lifted. Raising the 
piston tends to produce 
a vacuum in the cylin- 
der ; but the air in the 
receiver and connecting 
tube expands, lifts the 
valve ^, and fills the 
cylinder. Thus each 
down-and-up stroke of 
the piston results in the 
removal of a portion of 
the air in the receiver. After several strokes of the 
piston, the air in the receiver becomes rarefied to such an 
extent that its expansive force is no longer sufficient to 
lift valve i, and no further exhaustion can be produced. 

Some air pumps are so constructed that the valves are 
opened and closed automatically at the proper moment so 
that a greater degree of rarefaction can be reached. 

154. The Condensing Pump. — The condensing pump is 
used to compress illuminating gases in cylinders for use in 
lighting vehicles, stereopticons, etc., and further for inflat- 
ing pneumatic tires, operating drills in mines, air-brakes 
on railway cars, and for many other purposes. The most 
common condensing pump is that used for inflating bicycle 




Fig. 119. — Air Pump. 



MECHANICS OF GASES 



157 




Fig. 120. — Inflating a Tire Thy Means 
of a Condensing Pump P. 



tires. Figure 120 shows the construction of such a pump. 
When the piston P is forced down, the air in the cylinder 
of the pump is driven into 
an air-tight rubber bag 
within the tire T. The 
small valve s opens to ad- 
mit the air into the tire, 
but closes to prevent its 
return. On lifting the 
piston a partial vacuum is 
produced in the cylinder, 
and the air from outside 
finds its way into the cyl- 
inder past the soft cup-shaped piece of leather attached to 
the piston. During the downward stroke of the piston, 
however, this leather is pressed firmly against the sides 

of the cylinder and thus prevents 
the escape of the air. Repeated 
strokes of the piston add to the 
mass of air already in the tire, and 
the process of pumping may be con- 
tinued until the tire is sufficiently 
inflated.* 

155. The Common Lift Pump. — 
The simplest pump for raising water 
from wells is the common lift pump. 
This consists of a barrel, or cylinder, 
(7, Fig. 121, connected with a well 
or other source of water by a pipe 
B. The entrance of this pipe into 
the cylinder is covered by a valve 
s, opening upward. In the cylinder 
is a closely fitting^ piston P which 

121. — Common Lift . / , f ^ , , 

Pump. can be raised or lowered by means 




Fig 



158 



A HIGH SCHOOL COURSE IN PHYSICS 



of a rod which is usually connected with a lever for con- 
venience in operating. The piston contains a valve ^, also 
opening upward. The action of the pump is as follows : 
When the piston is raised, the pressure of the air in the 
tube and lower part of the cylinder is diminished. The 
atmospheric pressure on the surface of the water in the 
well then forces water into the pipe. As the piston is 
lowered, valve s closes, and t allows the air in the cylinder 
to escape, but closes again when the piston begins to ascend. 
The second stroke again reduces the pressure below the 
piston, and water is forced still higher in the pipe. At 
last the water reaches the piston, passes through during 
the downward stroke, and is lifted toward the spout of the 
pump when the piston moves upward. 

It will be seen that the action of the common lift pump 
is dependent on the pressure of the atmosphere to elevate 
the water to a point just high enough to come within reach 
of the piston ; the piston must not he farther above the water 

in the well than the height of 
the water column that the air 
can support. When water 
is to be lifted from a well 
more than 33 or 34 feet in 
depth, the cylinder is placed 
low enough to enable the 
piston to move within that 
distance of the surface of 
the water. 

156. The Force Pump. — 
The force pump is used to 
deliver water under consid- 
erable pressure either for 
spraying purposes or in or- 
FiG. 122. — A Force Pump. ^cr to elevate it to a reser- 




MECHANICS OF GASES 159 

voir placed some distance above the level of the pump. It 
is made in many different forms. Usually the force pump 
has no opening or valve in the piston. The water escapes 
from the cylinder through a side opening A^ Fig. 122, past 
the valve t, thence upward to the spout or reservoir. In 
other respects the description of the action of the lift pump 
applies equally well to the force pump. In order to obtain 
a steady stream of water, a force pump is often provided 
with an air chamber D. The entrance of the water through 
t serves to compress the air in the chamber, which by its 
expansive force maintains the current in pipe U while the 
piston is moving upward. 

157. The Siphon. — The siphon is a bent tube with un- 
equal arms. It is used for removing liquids from tanks or 

reservoirs that have no outlet, or for ^_^^ ^^ p-ca 

drawing off the liquid from a vessel ^ y ^ 

without disturbing a sediment lying 
upon the bottom. 

Let a glass tube bent in the form shown in 
Fig. 123 be filled with water, and the ends 
closed while the tube is inverted and placed 
in the position shown. On opening the ends 
water will flow through the tube from the 
vessel in which the liquid has the higher 
level. 

The action of the siphon is explained wWm 

as follows : the upward pressure at a iFflL 

due to the atmosphere is only partly 
counterbalanced by the pressure of the ^^^- 123. -The Siphon, 
liquid column ab. If the atmospheric pressure is p, the 
resultant force acting toward the right is p — ah. Again, 
the upward pressure of the atmosphere at d is opposed 
by the pressure due to the liquid column cd. Hence 
the resultant pressure acting toward the left is p — cd. 



"MmHiiiiiiiiiiiinnTTiTTTirl 



160 A HIGH SCHOOL COURSE IN PHYSICS 

Since the pressure due to the liquid in the long arm 
exceeds that of the liquid in the short arm, the excess 
of force in the tube tends to move the water toward 
the lower vessel. The resultant of all the forces is 
equal to the difference of pressure due to the liquids in 
the two arms, i.e. to. a column of liquid equal to cd — ah. 
The liquid will therefore continue to flow until this dif- 
ference is ; or, in other words, until the liquid reaches 
the same level on the two sides. A second condition 
necessary to the action of a siphon is that for water the 
height of the hend f must not he greater than 33 or S4:feet 
ahove a, since its elevation to the point / is due to the 
atmospheric pressure. 

158. Work done by Compressed Gases. — Since a gas 
exerts a force against the sides of the vessel containing it, 
it is obvious that work will be done by the gas if the side 
of the vessel is made movable and the gas allowed to ex- 
pand (§ 55). Imagine air to be 



T - ^p — > compressed in the cylinder O shown 

i w^^ in Fig. 124, in which P represents 



Fig. 124. — An Expanding a movable piston. If the external 

pfston P. '*'''' ^'''^°'' resistance offered to the piston is 

not as great as the force exerted 
by the air inside, work will be done by the expanding 
gas, and the piston will move. If the pressure remains 
constant during the process, the work done will be 
measured by the product of the force exerted by the air 
against the piston and the displaeement produced. 

Example. — A tube leads from a cylinder to a reservoir where air is 
stored under a pressure of 5000 g. per cm. 2; the area of the piston is 50 
cm. 2. Find the work done by the gas when the piston moves 20 cm. 

Solution. — The total force exerted by the gas against the piston 
is 50 X 5000, or 250,000 g. Hence the work done is 250,000 x 20, or 
5,000,000 grain-centimeters. 



MECHANICS OF GASES 



161 



Since energy can be transferred from place to place in 
a compressed gas, air under great pressure is often con- 
ducted through pipes over long distances from a condens- 
ing apparatus (§ 154) to a point where the energy is to be 
utilized. In this way a locomotive engineer is able to con- 
trol a train by means of compressed air conducted from 
the engine, where it is compressed, to suitable apparatus 
in connection with the brakes under each car (§ 159). 
In mining operations and stone quarrying, pneumatic ma- 
chines utilize compressed air for drilling the holes in rocks 
where explosives used in blasting are to be placed. Many 
other applications of the power of compressed gases have 
found a place in modern engineering. 




Trai'/t J-*ipe 




159. The Air Brake. — Compressed air is widely used by the 
railroads of many countries in the operation of the Westinghouse air 
brake, Fig. 125, which works as follows : — The locomotive is provided 
with a condensing pump 
which keeps the air in a 
large reservoir at a pres- 
sure of about 80 pounds 
per square inch. From 
this reservoir the com- 
pressed air is conducted 
through the train pipe P 
to an auxiliary reservoir jr^ ^J^^ 
R placed under each car. Brake, 
As long as the pressure is 
maintained in P, air is p^^ ^i^ 
allowed to enter R and at 

the same time is prevented from entering the cylinder C by a compli- 
cated automatic valve F, and the brakes are held " off " by the spring 
S. If, however, the engineer by moving a lever in the cab, allows the 
pressure in the pipe P to fall, the passage between R and P is at 
once closed by the valve F, and the compressed air in R is admitted 
into C The pressure of the air forces the piston to the left and 
sets the brakes against the wheels. By admitting air from the reser- 
voir on the engine into the pipe P, the valve F again establishes a 
12 




^W!V^W;^^^ ^^'^^ ^^ 



■The Westiuohuuse Air Brake. 



162 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 126. 



communicatiou between P and R and allows the air in C to escape. 

The sjjring S forces the piston back and releases the brakes. The 

great advantage of this brake is 
that in case of any accidental 
breaking of the train pipe P, 
the brakes are automatically set. 
They are often arranged to be 
operated from any coach in case 
of emergency. 

160. Subaqueous Opera- 
tions. — Important use is made 
of compressed air in various en- 
gineering operations performed 
under water, as laying the foun- 
dations of bridges, excavating for 
tunnels, recovering the cargoes 
from sunken vessels, etc. Figure 
126 shows the most important 
apparatus for subaqueous work, 
the diving bell, and the sub- 
marine diver. A condensing 

pump on board the vessel forces air through a tube into the bell, 

thus supplying the workman with oxygen and preventing the rise 

of water in the working chamber. 

Air is supplied to the workman 

in diving armor in the same 

manner. The foul air escapes 

from the diving suit through a 

valve above the chest. In many 

cases the air is supplied from a 

reservoir carried on the back of 

the diver. 

Deep excavations are fre- 
quently made and foundations 

built up from bed rock by the 

aid of the pneumatic caisson 

(pronounced kds'son). See Fig. 

127. The working chamber C BedBock 

is gradually lowered through soft ^^^ 127. -Excavatiug below the Water 

soil by removing the earth from Level by the Aid of Compresseil Air. 



Apparatus Used for Work 
under Water. 




MECHANICS OF GASES 163 

within after loading the roof of the chamber with heavy masonry 
above. AVhen the caisson sinks below water level, air under suitable 
pressure is forced into C to prevent the influx of water. The ex- 
cavated earth is removed, and the foundation material introduced 
through the air lock A. The bucket is lowered into A, after which 
the opening around the wire cable is closed air-tight. Air is then 
allowed to flow into A until the pressure there is equal to that in C. 
The semicircular doors separating A and C are now thrown open, and 
the bucket lowered into the caisson. As the caisson is forced more 
and more below water level, the pressure of the air is correspondingly 
increased in order to prevent the water and mud from crowding in. 

EXERCISES 

1. If the pressure against the 8-inch piston of an air brake is 75 lb. 
per square inch, how much force drives the piston forward? 

2. When a train is broken in two, the cars are brought quickly to 
rest. Explain. 

3. What are the advantages gained by the use of air brakes? 

4. A diver sinks 68 ft. below the surface of water. Under how 
many atmospheres is he working? Explain w^hy his body is not 
crushed by this force. 

5. A caisson is sunk until the bottom is 51 ft. below water level. 
Under what pressure must the laborers work ? 

6. Explain how a bucket of earth is removed from a caisson through 
the air lock. 

SUMMARY 

1. The laws of pressure relative to liquids are equally 
applicable to gases, except that the pressure is not propor- 
tional to the depth (§§ 135 and 136). 

2. The density of the air under standard conditions of 
pressure and temperature (i.e. 76 cm. and 0° C.) is 1.293 
g. per liter or about 1.25 oz. per cubic foot (§ 137). 

3. The pressure of the atmosphere is measured by the 
barometer. At sea level the average height of the mer- 
curial column supported by the air is 76 cm. or very nearly 
30 in. Expressed in units of force, the average sea-level 
pressure is 1033.6 g. per square centimeter or 14.7 lb. per 



164 A HIGH SCHOOL COURSE IN PHYSICS 

square inch. The atmospheric pressure decreases with 
the altitude, but not proportionally (§§ 139 to 146). 

4. Gases adapt themselves to the form and capacity of 
the vessels containing them. The expansive force of a gas 
confined in a receptacle tends to prevent the collapse of the 
vessel under atmospheric pressure (§ 147). 

5. The product of the pressure and volume of a given 
mass of a gas at a constant temperature is constant. This 
is known as Boyle s Law^ or Mariotte's Law (§149). 

6. The density of a gas at a constant temperature is 
directly proportional to the pressure to which it is subjected 

(§ 150). 

7. A body in air is buoyed up by a force equal to the 
weight of the air that it displaces (§ 152), 

8. The air pump is used in the rarefaction of gases. 
The condensing pump is used in compressing air and 
other gases (§§ 153 and 154)* 

9. The action of " suction " pumps is dependent upon 
the pressure of the atmosphere on the water in the well or 
cistern (§§ 155 and 156). 

10. The siphon is a bent tube having unequal arms 
used for conveying a liquid over the side of a reservoir to 
a lower level than that in the reservoir. Its action is 
dependent on atmospheric pressure (§ 157). 

11. Compressed air possesses energy. This energy is 
used in many important mechanical devices, as the air 
brake, rock drills, etc. (§ 158). 



CHAPTER IX 



SOUND : ITS NATURE AND PROPAGATION 



1. ORIGIN AND TRANSMISSION OF SOUND 

161. Cause of Sound. — Whenever the sensation of sound 
is traced to its external cause, we find that its source is 
always something which is in a state 
of vibration. Sometimes the vibra- 
tion of the body emitting the sound 
is sufficiently great to be visible, i.e. 
to give a certain blurred indistinct- 
ness to the outline of the body. This 
is easily perceived when a stretched 
wire is plucked or a tuning fork is 
sounded. 





Fig. 128. — Demonstrating 
the Motion in a Sound- 
ing Tuning Fork. 



Let a small pith ball suspended on a thread 

be allowed to touch a tuning fork that is 

emitting a sound. (See Fig. 128.) It will be thrown violently away. 

Touch one of the prongs lightly to the water in a tumbler. A ripple 

is produced, or perhaps a 
spray is thrown from the 
prong. Again, attach a 
fine wire or bristle to the 
prong of a tuning fork with 
a small quantity of sealing 
w^ax. Sound the fork, and 
draw the bristle across a 
piece of smoked glass. A 
wavy line, Fig. 129, results, 

showing the existence of a hach-and-forth motion in the fork. 

162. The Nature of a Vibration. — The vibration of a 
pendulum has been studied in § 80. While sounding 

165 




Fig. 129. —The Motion in the Prong of a Fork 
is Vibratory. 



166 



A HIGH SCHOOL COURSE IN PHYSICS 



bodies vibrate in a manner similar to that of the pendu- 
lum, they vibrate from a different cause. When a pendu- 
lum is drawn aside and then set free, a component of the 
force of gravity moves the pendulum bob back toward its 
original position at the center of its arc. When, how- 
ever, we pluck the string of a guitar, for example, or set 
the prongs of a tuning fork in vibration, the motion is due 
to the elasticity of the material of which the body is com- 
posed. The string, on being draw^n aside, is stretched 
slightly. Now, on being released, its tendency to resume 
its original length causes it to straighten. When we set 
a tuning fork in vibration, the prongs are bent. Since 
they are made of elastic steel, they immediately tend to 
resume their original shape. Hence the prongs move 
back to their initial positions. Again, the string of the 
guitar, like the pendulum, is in the state of motion when 
it reaches its original position. Hence it must continue 
to move until some resistance checks it. Therefore it 
swings beyond this position to a point where its velocity 

is zero, whence it returns in the 
same manner as before. The phe- 
nomenon is repeated by the string 
until, like the pendulum again, its 
energy is expended in overcoming 
the resistance of the air and the 
friction of its own molecules. 




Clamp a thin strip of wood, as a yard- 
stick, by one end, Fig. 130, and set it in 
vibration. As it is being drawn aside, its 
tendency to move back toward its original 
position is very apparent. Let the stick 
move very slowly back to the original posi- 
tion, and it will stop there ; but if it is set 
entirely free, it will continue to move until it is some distance beyond 
the center. Why ? Attach a weight of 100 or 200 grams near the 



Fig. 130. — Thin Strip of 
Wood in Vibration. 



SOUND: ITS NATURE AND PROPAGATION 167 



free end of the stick, and it will be found to vibrate much more slowly 
than before. The force due to the elasticity of the wood cannot bring 
the increased mass so quickly to the center nor stop it so quickly 
when the center is passed on account of the increased inertia. Make 
a comparison between the vibrating yardstick and the sounding tuning 
fork. The yardstick does not vibrate with sufficient rapidity to pro- 
duce sound. 

163. Transmission of Sounds to the Ear. — Sounds reach 
the ear through the air as the transmitting medium. 
Many other substances may, however, be the means of 
propagation, as the following experiments will show : 

1. Let the ear be held against one end of a long bar of wood, and 
let the shank of a small vibrating tuning fork be brought against the 
other end. A loud sound will be heard. The scratch of a pin at one 
end can easily be heard at the other. 

2. Place the shank of a tuning forR in a hole bored in a large cork. 
Set a tumbler full of water upon a resonance box, and bring the cork 
in contact with the surface of the water. If the fork is in vibration, 
the water will transmit the motion to the box, as shown by the in- 
creased intensity of the tone. Again, as most boys have found experi- 
mentally, if the ear be held beneath the surface when two stones are 
struck together under water, a loud sound results even at some dis- 
tance from the stones. 

164. Medium Necessary for Propagation of Sound. — That 
a sounding body cannot be heard without the presence of 
some transmitting medium may be shown 
by the aid of the air pump. 



Let an electric bell be placed upon a thick pad of 
felt or cotton, suspended by flexible wire springs 
under the receiver of an air pump, as shown in Fig. 
131. Make the connection with a battery, so that 
the bell can be rung from the outside. Set the bell 
ringing, and begin to exhaust the air from the re- 
ceiver. The sounds coming from the bell become 
less and less distinct until the greatest possible ex- 
haustion has been produced. If now the air is slowly 




Topump 

Fig. 131.— Bell 
Ringing in a 
Partial Vac- 
uum. 



168 A HIGH SCHOOL COURSE IN PHYSICS 

admitted into the receiver, the loudness of the sound increases until 
its full intensity is reached. 

Although the sound of the bell used in this experiment 
will never become entirely inaudible, mainly on account 
of the transmission of sound by the vibration of the sup- 
porting wires, we are given reason to believe that with com- 
plete exhaustion and the removal of all other transmitting 
media, no sound would be heard. 

165. Velocity of Sound Transmission. — Every one is 
familiar with the fact that it requires time for sound to 
travel over a given distance. If we watch a locomotive 
from a distant point as the engineer blows the whistle, we 
first observe the jet of steam as it issues from the whistle, 
and a few seconds later we hear the sound. The interval 
of time that often elapses Jbetween a lightning flash and 
the peal of thunder is further evidence that the velocity 
of sound is not exceedingly great. 

Many investigators during the nineteenth century gave 
their attention to the accurate determination of the ve- 
locity of sound. For the most part their experiments 
consisted in measuring the interval of time between the 
flash of a gun and its report heard *at some distant sta- 
tion of observation. The result obtained by Regnault, a 
French physicist, gives sound a velocity, in air, of 1085 feet 
per second at the temperature of freezing water. This ve- 
locity is equivalent to 331 meters per second. At higher 
temperatures the velocity is somewhat greater, the in- 
crease being 2 feet, or 0.6 meter, per second for each de- 
gree centigrade. 

166. Velocity of Sound in Various Mediums. — The 
velocity of sound in solids has been the subject of many 
investigations. The accepted value found for iron is 
about 5100 meters per second at 20° C. The velocity of 
sound in wood depends greatly upon the kind of wood. 



SOUND: ITS NATURE AND PROPAGATION 169 

The average value, however, is approximately 4000 meters 
per second. 

The most exact measurement of the velocity of sound in 
water was made in 1827 by Colladon and Sturm in Lake 
Geneva, Switzerland. Two observers stationed them- 
selves in boats at opposite sides of the lake. At one of 
the stations a bell was sounded beneath the water and a 
gun fired on deck at precisely the same instant. The 
sound was received by means of a large ear trumpet held 
under water at the other station. The time intervening 
between the stroke of the bell and the report transmitted 
by the water could thus be ascertained and the velocity 
computed. The results of many observations gave an 
average of 1400 meters per second as the velocity of sound 
through water. 

EXERCISES 

1. The flash of a gun is seen 3.5 seconds before the report is heard. 
If the temperature is 20° C, what is the distance between the observer 
and the gun ? 

2. A locomotive whistle was sounded 3 mi. from an observer. If 
the temperature of the air was 10^ C, how long was the sound in 
traversing the distance ? 

3. The distance between two stations is 12 mi. If the interval of 
time between the flash and the report of a gun was found by experi- 
ment to be 56 seconds, what was the speed of the sound ? 

4. A bullet was fired at a target 500 m. away, and in 3 seconds 
was heard by the gunner to strike. The temperature of the air being 
20"^ C, what was the velocity of the bullet? 

5. When one end of an iron pipe is struck a blow with a hammer, 
an observer at the other end hears two sounds, one transmitted by the 
iron, the other by the air. If the pipe is 1500 m. long, and the tem- 
perature 25° C, what is the interval of time between the two sounds ? 

2. NATURE OF SOUND 

167. Sound a Wave Motion. — We have seen in § 164 
that a medium is essential for the transmission of sound, 



170 A HIGH SCHOOL COURSE IN PHYSICS 

but thus far no explanation has been given of the manner 
in which this transmission takes place. Souuds continue 
to come from an electric bell even though we cover it 
tightly with a glass jar, but it is very plain that nothing, 
i.e. no material thing, can pass through the glass. The 
whole process will become clear, however, if we consider 
that a sound is transmitted through the air and glass in the 
form of waves. Hence, the further study of Sound will be 
a study of waves and wave motion. 

168. Two Kinds of Waves. — We are all familiar with 
the waves that move over a surface of water. We have 
only to observe a small boat as it rises upon the crests and 
sinks down into the troughs to realize that it is not carried 
along by the wave. Again, a wave is often seen to pass 
over a field of grain. While the wave moves rapidly 
across the field, each spear of grain simply bends with the 
pressure of the wind and then rises again. The following 
experiment may be used to show this manner of wave 
propagation : 

Let one end of a soft cotton clothesline about 25 feet long be 
attached to a hook in the wall, while the other end is held in the 
hand. Give the end of the rope a quick up-and-down motion, and a 
wave will be seen to run along the rope from one end to the other. 

In the experiment just described it is clear that the for- 
ward motion of the waves produced in the rope is at right 
angles to the direction of the motion of the particles compos- 
ing the rope, as shown in 
Fig. 132, On this ac- 
count such waves are 
Motion of ffiepat^ic^es Called transvcrsc waves. 

Fig. 132. —Transverse Waves in a Rope. ^ second kind of wave 

motion takes place in bodies that are elastic and com- 
pressible, as in gases, wire springs, etc. We are able to 
make a study of waves of this kind by letting a coil of 




SOUND: ITS NATURE AND PROPAGATION 171 

wire, Fig. 133, represent the medium through which such 
waves are transmitted : 

Let us imagine a blow is given the spring at A that 
quickly compresses a few turns of the spiral near the end. 
On account of the elasticity 
and inertia of these parts of 
the springy, they will move 

^ ^' '^ Fig. 133. — Waves Transmitted by a 

forward slightly and com- Spring. The individual Turns 

press those just ahead. Move back and forth while the 

^ _ *• _ Waves Progress along the Spring. 

These in turn will compress 

the coils still farther along, and thus a pulse, or wave, 

is carried along the spring from A to B. 

Again let the end of the spring at A be given a very 
sudden pull. The turns near the end will be drawn apart 
for an instant, but the adjacent turns will be drawn toward 
A^ one after another^ until the end B receives the impulse. 
A blow against A is transmitted by the spring to the point 
B, and a sudden pull at A is likewise transmitted as a 
pull against the fastening at B, Waves of this kind, in 
which the motion of the parts of the medium are parallel 
to the direction of propagation, are called longitudinal 
waves. 

169. Transmission of a Sound Wave. — The manner in 
which a sound is transmitted by the air becomes clear 
when we compare the process with that which occurs 
when a wave is transmitted by a spiral spring. 

Imagine a light spring, (1), Fig. 134, to be attached 
at one end to one of the prongs of a vibrating tuning 
fork F and at the opposite end to a diaphragm Cr. Each 
vibration of the fork will alternately compress and separate 
the spirals of the spring near the end. These pulses will 
be transmitted by the spring in the manner described in 
§ 168, and will cause the diaphragm at G- to execute as 
many vibrations per second as the tuning fork, and the 



172 



A HIGH SCHOOL COURSE IN PHYSICS 



diaphragm will give out a sound corresponding to that 
emitted by the fork. 

Now let the air take the place of the spring and the 
ear U replace the diaphragm. When the prong of the 
vibrating fork, (2), Fig. 134, moves to the right, a com- 
pression of the air is produced in front of it. This com- 
pression, or condensation, moves to the right with the 
velocity of sound, or at about the rate of about 1120 feet 
per second. When the prong of the vibrating fork moves 
to the left, the air just at the right is rarefied, and the ad- 
jacent portions of the air move in to fill this rarefaction, 




Fig. 134. — Illustrating the Corresponding Parts of Transverse 
and Longitudinal Waves. 

which travels to the right immediately following the con- 
densation. Condensation and rarefaction thus follow one 
another as long as the fork continues to vibrate, and the 
drum of the ear at E^ like the diaphragm in the preceding 
case, receives as many pulses per second as the tuning fork 
emits. It is therefore caused to vibrate as many times per 
second as the fork. 

The curve in (3), Fig. 134, is drawn to show the parts of 
a transverse wave which is sometimes used to represent 
diagrammatically waves of other kinds. The crest AB 
corresponds to the condensation^ or compression, of the 
longitudinal waves shown au (1) and (2). The trough 



SOUND: ITS NATURE AND PROPAGATION 173 



BC corresponds to the rarefaction. The distance ah rep- 
resents the distance that eacii particle in the wave moves 
from its original position and is called the amplitude of the 
wave. A wave includes a complete crest and trough, or a 
condensation and a rarefaction. The distance between the 
corresponding points on any two adjacent waves, as AC, 
BD, etc., is the wave length. 

170. Velocity, Wave Length, and Vibration Frequency. — 
Imagine a tuning fork whose rate of vibration, or vibration 
frequency., is 256 per second. When the fork is set in vi- 
bration, it sends out 256 complete longitudinal waves dur- 
ing each second. At the completion of the 256th vibra- 
tion, the first wave has progressed a distance numerically 
equal to the velocity of sound, or about 1120 feet. This 
space, therefore, contains 256 waves, and the length of each 
wave can be found by dividing 1120 feet by 256. Hence 
the length of each wave is about 4.4 feet. Letting n be 
the frequency, v the velocity of sound, and I the wave 
length, we have 



n 



or nl = v. 



(1) 



3. INTENSITY OF SOUND 

171. Sound Waves Spread in all Directions. — If a bell 
be struck, the sound is heard as readily in one direction 
as another if no ob- 
struction intervenes 
and other conditions 
are equally favorable. 
Ob^-iously a wave 

emitted by the bell Fig. 135.— Soundwaves Spread in all Directions 
spreads out in all from their Source. 

directions from it. Figure 135 illustrates this fact. Each 
condensation has the form of a hollow spherical shell that 




174 A HIGH SCHOOL COURSE IN PHYSICS 

continually enlarges as the wave advances. This is 
followed by a rarefaction of a similar forn:i. It is plain 
that the energy imparted by the bell to a single wave is 
carried from the source by particles of air composing a 
hollow spherical shell. At a given distance from the bell 
this shell will have a certain area which at twice the dis- 
tance will be four times as great, since the area of a 
sphere is directly as the square of its radius. Thus at 
twice the distance from the bell the energy will be im- 
parted to four times as many particles, hence the energy 
of each will be one fourth as great. At three times 
the original distance, the energy will be imparted to nine 
times as many particles, and the energy of each will be 
one ninth as much. Hence, the inteyisity of sound is in- 
versely proportional to the square of the distance measured 
from the source of the sound. 

172. Intensity and Amplitude. — The energy imparted 
to each wave by a sounding body will depend upon the 
intensity of the vibration of that body, i.e. upon the am- 
plitude of vibration. An increase in the amplitude of 
vibration of the sounding body produces a corresponding 
increase in the amplitude of each wave emitted. Thus 
each wave is caused to carry away from the sounding 
body an increased quantity of energy. When the waves 
fall upon the tympanum of the ear, a corresponding in- 
crease in its amplitude of vibration is produced. 

Let one of the prongs of a tuning fork be struck lightly against a 
soft pad or cork. A tone is emitted that is scarcely audible. Now 
let the fork be struck more forcibly. On account of the greater am- 
plitude of vibration the intensity of the tone will be much greater 
than before. 

Like the pendulum (§§ T9 and 80), a sounding body 
completes its vibrations in equal times, the period of vi- 
bration being practically independent of the amplitude 



SOUND: ITS NATURE AND PROPAGATION 175 

(i.e. isochronous^. Therefore when the fork is struck 
forcibly against the pad, it still performs the same number 
of vibrations per second as before. 

173. Intensity and Density of the Medium. — The ex- 
periment with the bell and air pump (§ 164) shows that 
the intensity/ of a sound depends upon the density of the 
medium in which the sound is pi^oduced. As the air in the 
receiver becomes rarer, the intensity of the sound grows less. 

Support two similar bell jars with the mouths opeuing downward. 
Keep one filled with illuminating gas and the other with air. Set an 
electric bell ringing, and place it first in the jar containing gas, then 
in the one filled with air. It will be readily observed that the sounds 
emitted in the rarer medium, the illuminating gas, are less intense 
than those produced in the jar filled with air. 

At the top of a high mountain a gun makes only a 
small report when lired. At high elevations explorers 
and aeronauts converse with difficulty. The denser the 
gas, the greater the energj^ imparted to each wave by the 
vibrating body. For this reason a sounding body will 
cease to vibrate sooner in the denser of two media. 

174. Intensity and Area. — When the area of a sound- 
ing body is small, as that of a small tuning fork, for 
example, the condensations and rarefactions produced are 
not well marked. It is obvious that of two vibrating 
tuning forks of the same frequency, the smaller will have 
the effect of cutting through the air without producing 
more than a slight compression, while the one of larger 
area will compress a greater volume of air, and thus give 
out more of its energy to each wave emitted. 

Set a tuning fork in vibration, and hold the shank against the 
panel of the door or table. A loud sound will be heard coming from 
the large area that is set in vibration by the fork. 

In the construction of many musical instruments use is 
made of this relation between the intensity of sound and 



176 



A HIGH SCHOOL COURSE IN PHYSICS 



the area of the sounding body. Thus a piano utilizes two 
or three wires to produce a given tone when the area of 
a single wire is not sufficiently great. Again, a sounding 
board of large area is often placed beneath the strings of 
instruments to form a large surface of vibration when the 
strings are excited. 



4. REFLECTION OF SOUND 

175. Reflection of Sound Waves. — It is a fact frequently 
illustrated in nature that sound waves may be reflected. 
As long as waves proceeding from a source of sound pass 
through a homogeneous medium, no reflection takes 
place ; but when the density of the medium is disturbed, 
the waves suffer partial or total reflection. 

Let a watch be placed a few 
inches iu front of a concave re- 
flector, as shown in Fig. 136. By 
moving the ear from point to point 
a place may be found, sometimes 
several feet from the reflector, 
where the sound of the watch may 
be distinctly heard. 

The experiment may be varied 

by setting a similar reflector at a distance of several feet from the 

first and facing it, as shown in Fig. 137. If the watch is placed 

slightly nearer the reflector than in the preceding experiment, a point 

may be found a few inches in front 

of the second at which the sound 

is focused. This point, or focus, 

may be found by using an ear 

trumpet made by attaching a piece 

of rubber tubing to a glass funnel. 

When the open funnel is placed at Fig. 137. — Sound Undergoes Two Ke- 

the focus of sound, a distinct tick- *^^^^^«^« *^°"^ ^^^^^^^ Surfaces. 

ing of the watch will be heard by holding the end of the rubber 

tubing in the ear. 




Fig. 136. 



Sound Reflected by a Con- 
cave Surface. 





SOUND: ITS NATURE AND PROPAGATION 177 

176. Echoes. — The familiar phenomenon of echoes is 
due to the reflection of sound. When one speaks in a 
room of moderate size, the waves reflected from the walls 
reach the ear so quickly that they combine with the direct 
waves and produce an increase of the intensity of the sound. 
But when the walls are one hundred feet or more from 
the speaker, he hears a distinct echo of each syllable he 
utters. Sometimes a reflecting surface is so distant that 
several seconds may intervene before an echo returns. 
Between two reflecting surfaces the echoes sent back and 
forth are often remarkable. It is related that at a point 
in Oxford County, England, an echo repeats a sound from 
fifteen to twenty times. Extraordinary echoes are also 
found to occur between the walls of deep canons. 

In so-called " whispering galleries " we have illustrated 
the formation of a sound focus due to curved surfaces. In 
the crypt of the Pantheon in Paris there is a place where 
the slight clapping of hands at one point gives rise to 
sounds of great intensity at another. In the Mormon 
Tabernacle at Salt Lake City, Utah, a whisper near one 
end can be distinctly heard at the other, so perfect is the 
reflection from the ellipsoidal surfaces of the walls. 

EXERCISES 

1. A hunter fires a gun and hears the echo in 5 seconds. How far 
away is the reflecting surface, the temperature being 20° C? 

2. How far is a person from the wall of a building, if, on speaking 
a syllable, he hears the echo in 4 seconds, the temperature being 
15° C? 

3. A gun is fired, and the echo is returned to the gunner from a 
cliff 250 ft. away in 4.5 seconds. Calculate the velocity of sound. 

SUMMARY 

1. Sounds are produced by bodies in a state of rapid 
vibration (§ 161). 
'^ 13 



178 A HIGH SCHOOL COURSE IN PHYSICS 

2. Sounding bodies vibrate as a consequence of the elas- 
ticity and inertia of the material composing them (§ 162). 

3. Sounds are transmitted from their sources by solids, 
liquids, and gases. The air is the propagating medium in 
most cases (§§ 163 and 164). 

4. The velocity of sound waves in air at the freezing 
temperature is 1085 ft. (331 m.) per second. The velocity 
is increased 2 ft. (0.6 m.) per second for each degree 
centigrade. In wood and iron the velocity of sound is 
respectively 3 and 4 (approximately) times the velocity 
in air (§§ 165 and 166). 

5. Sound is a wave motion. Wave motion may be 
either transverse or longitudinal (§§ 167 and 168). 

6. The energy of a sounding body is transmitted by 
longitudinal waves. The parts of such, waves are called 
condensations and rarefactions. The corresponding parts 
of transverse waves are crests and troughs (§ 169). 

7. The relation between velocity, wave length, and 
number of vibrations per second is given by the equation 
z; = Zw(§170). 

8. The intensity of a sound depends on the amplitude 
and area of the sounding body, the density of the medium 
where the sound is produced, and is inversely proportional 
to the square of the distance from its source (§§ 171 to 174). 

9. Echoes are sounds reflected from walls, woods, hills, 
cliffs, etc. (§§ 175 and 176). 



CHAPTER X 



SOUND : WAVE FREQUENCY AND WAVE FORM 

1. PITCH OF TONES 

177. Musical Sounds and Noises. — In order that a 
sound may be pleasing to the ear, it is essential that the 
vibrations be made in precisely equal intervals of time ; 
in other words, the vibrations must be isochronous. Any 
device that produces isochronous pulses emits a musical 
sound. If the pulses are not isochronous, the result is a 
noise. 

Rotate on a whirling table a metal or cardboard disk provided with 
two or more circular rows of holes, Fig. 138. Let the holes in one row 
be equidistant, while those in the other rows 
are placed at irregular intervals. Blow a 
stream of air through a rubber tube against 
the row of equidistant holes, and a pleasant 
musical sound will result. Now direct the 
stream against another row of holes, and 
it will be observed that the sound produced 
is of an unpleasant character. 

178. Pitch. — Probably the most 
striking difference in musical sounds 
is in respect to that which we call 
pitch, a term applied to the degree of 
highness or lowness of a sound. We 
are accustomed to sounds varying in pitch from the low, 
rumbling thunder to the shrill, piercing creak of small 
animals and insects. How this wide difference in sounds is 
brought about may be shown by the following experiment: 

179 




Fig. 138. — Musical Sounds 
Produced by Isochro- 
nous Pulses in the Air. 




180 A HIGH SCHOOL COURSE IN PHYSICS 

Rotate a metal or cardboard disk in which there are several circu- 
lar rows of holes perforated at regular intervals, but having a different 
number in each row. See Fig. 139. Keeping the 
speed of rotation constant, blow a stream of air 
forcibly against one row and then another. A dis- 
tinct difference in pitch will be observed. Again, 
keeping the stream of air directed against one of 
the rows, change the speed of rotation from very 

Fig 139 The ^^^^ ^^ ^^^^ ^^^^' "^^^ pitch \yll rise from a low 

Siren Disk. tone to one that is very shrill. 

These facts teach that the pitch of a tone depends upon 
the number of wave pulses per second sent from a sounding 
body to the ear. The stream of air blown against the disk 
is alternately transmitted and interrupted by the motion 
of the holes. The succession of pulses thus produced 
constitutes the musical tone that is heard. The vibration 
frequency of the tone emitted may be found by multiply- 
ing the number of revolutions of the disk per second by 
the number of holes in the circle. An instrument of the 
kind used in the experiment is called a siren, 

179. Musical Scales Produced by a Siren. — The most 
common series of tones of different pitches is the musical 
scale. It is possible with the help of a siren designed in 
a manner similar to the- one shown in Fig. 139 to deter- 
mine the relation of the several tones that comprise the 
ordinary musical scale. The construction and use of the 
siren is as follows : 

On a circular disk of cardboard or metal, about 10 inches in diame- 
ter, draw eight concentric circles about \ inch apart. Upon the circles 
thus drawn, beginning with the smallest, drill the following numbers 
of equidistant holes : 24, 27, 30, 32, 36, 40, 45, and 48. Mount the 
disk upon a whirling table and rotate with a uniform speed. Be- 
ginning with the smallest circle, blow a stream of air against each 
row of holes in succession. The tones produced will be recognized 
at once as those belonging to the major scale. Increase the speed of 
rotation, and again direct the current of air against the several rows 



SOUND: WAVE FREQUENCY AND WAVE FORM 181 

in succession. Although the pitches are higher than before, yet the 
scale is produced as perfectly as at first. 

The experiment teaches that the major scale is a series 
of tones whose vibration frequencies have the same rela- 
tion as the numbers 24, 27, 80, 32, 36, 40, 45, and 48. 
To these tones we give the names do^ re, wz, /a, sol^ la, 
ti, do. These tones form the foundation upon which has 
been built up our musical system. 

180. The Major Diatonic Scale. — Any series of tones 
whose vibration frequencies, bear the relations given in the 
preceding section constitute a major diatonic scale. The 
first tone of such a series is the key tone. The various 
scales in use are named according to their key tones ; for 
example, the scale of (7, the scale of (7, e^c. Physicists as- 
sign to the tone called middle (7, 256 vibrations per second. ^ 
Hence the second tone of the major scale of O must be 
produced by 256 x ||, or 288 vibrations per second. This 
tone is called D. The next tone, or E, must have 
256 X ||, or 320 vibrations per second, etc. The follow- 
ing table, Fig. 140, shows the manner in which these tones are 
expressed on the musical staff, their relative and absolute 
vibration numbers, and the vibration ratios. 





Staff 


[-5 1 




/_ 












^^ 


1 




i^\ ^ -r \ \ 


Scale 
of <7 


\ ) 






a V 


W 






1 1 




Lr 


-^ 


■ 9 
1 


V 1 
1 








1 




Letters 


C 


D 


f F 


G 


A 


B 


c 




[ Absolute Vib. No. 


25G 


2S8 


320 3kl.3 


381, 


U2G.G 


uso 


512 


For all 

Scales 


' Tone Numb 
Syllables 
Relative Vil 
Vibration R 


er 

). No. 
atios 


1 
do 

1 


2 
re 

27 


3 A 
mi fa 

30 32 
f t 


5 
sol 

36 


6 
la 

ho 

6 

3 


7 
ti 


8 
do 

U8 

2 



Fig. 140. The Major Scale. - . 

The nature of the seven definite intervals leading from 
C to C, or one octave, may be seen from the table. Start- 

1 The international standard of pitch in this country and Europe is 
based upon A = 435 vibrations per second. 



182 A HIGH SCHOOL COURSE IN PHYSICS 

ing with C = 256 vibrations, the vibration numbers of the 
successive tones of the scale, as shown by the experiment 
in the preceding section, bear the ratios |, J, f, |, f, 
^g^-, and 2 to the vibration number of (7, the key tone. 
These ratios are the same for all major scales^ no matter 
what the key tone may he. 

181. Intervals. — A musical interval refers to the relation 
between the pitches of two tones. The experiment with the 
siren in § 179 shows that the value of an interval depends 
upon the ratio^ not the difference., between the vibration 
numbers of the tones ; for example, when this ratio is 
IJ, or 2:1, the interval is an octave^ as the interval be- 
tween C and O. Other important intervals are the sixths 
i.e. the interval between the first tone and the sixth of a 
major scale, given by the ratio ||, or 5:3; the fifths |J, 
or 3:2; the fourth, ||, or 4:3; the major third., ||, or 
5:4; and the minor third., |^^, or 6 : 5. In order to determine 
the interval between two tones, the ratio of their vibration 
frequencies is first computed and then compared with the 
ratios given for the several cases. Hence two tones of 500 
and 300 vibrations per second respectively are a sixth apart, 
because the ratio of their vibration numbers is 5:3. If 
these two tones were sounded together or in succession, 
the interval would be recognized at once by a musician. 

182. The Major Chord. — A careful inspection of the 
vibration numbers of the tones of the major scale will 
show the presence of three groups consisting of three tones 
each, whose frequencies bear the ratios 4:5:6. Such a 
group of tones is a major chord. Beginning with C= 256 
vibrations per second, we have for the first chord (7, E., 
and G- (do-mi-sol) whose vibration frequencies are 256, 
320, and 384. This chord is called the tonic triad of the 
scale of C. The second of these chords, called the sub- 
dominant triads is formed in the same manner and includes 



SOUND: WAVE FREQUENCY AND WAVE FORM 183 



F^ A, and C (^fa-la-do^. The third is formed likewise 
by making use of the tones G-, P, and D' (^sol-ti-re) and 
is called the dominant triad. Since these triads include 
all the tones of the major scale, it may be said that this 
scale is founded upon these three major chords. The fol- 
lowing table shows these relations: 

Tonic Triad 4 : 5 : 6 : : 256 : 320 : 384, C, £, and G. 

Suhdominant Triad 4 : 5 : 6 : : 341 : 427 : 512, F, A, and C. 
Dominant Triad 4 : 5 : 6 : : 384 : 480 : 576, G, B, and D'. 

183. Sharps and Flats. — The introduction of the black 
keys on the organ or piano keyboard is (1) for the pur- 
pose of accommodating the instrument to the range of the 
voice in the case of songs and (2) to give variety to selec- 
tions designed for instrumental performance. That it is 
necessary to insert additional tones becomes apparent, at 
once when we consider the tones that are required to form 
a major scale beginning upon B^, just below middle (7, hav- 
ing 240 vibrations per second. The keys that are used in 
the scale of O are all white keys, Fig. 141, and have the 
frequencies indicated. ^_ 
In the major scale of ^ 
B the vibration fre- 
quencies must be suc- 
cessively 240, 270, 
300, 320, 360, 400, 450, 
and 480. It will be 
observed that the only ' 

white keys that satisfy Fig. 141 
this scale are E= 320 
and B = 480 vibrations per second. Since the number 270 
lies about midway between the frequencies of Q and i>, the 
black key (7* (read "(7 sharp") is introduced. Others must 
be placed between D and E, F and Gr, G- and A, and A and 
B, These are called respectively i>*, F^, Cr^, and A^, 



^^ ? T T ! r^ 

c\d]e f\g\a\b\c^] 



Illustrating the Scale of C on Staff 
and Keyboard. 



184 A HIGH SCHOOL COURSE IN PHYSICS 

These four tones are also called E^^ (r^, ^ , and B , 
and are read " E flat," " G flat," etc. 

184. Tempered Scales. — In the preceding section is 
shown the necessity for introducing additional tones in 
order that a piano, for example, may be used to produce 
the major scale of B. These new vibration numbers, how- 
ever, will not satisfy scales which begin on other key 
tones, for every change to another key tone increases the 
demand for small changes in the vibration numbers of the 
tones. The difficulty is surmounted by sacrificing the 
perfect, simple ratios found in §§ 180 to 183 and substi- 
tuting others that are sufficiently near to satisfy a musical 
ear. See the table given below. This method of tuning 
an instrument is called tempering, and the scales derived 
by the process are called tempered scales. The desired 
result^ which is to permit the execution of musical compo- 
sitions written in any key^ is secured hy making the inter- 
val between any two adjacent tones the same throughout the 
length of the keyboard. By this process the octave is di- 
vided into twelve precisely equal intervals called half 
steps. The imperfection introduced by equal tempera- 
ment tuning is illustrated by the following table : 

CDEFGABC 

Perfect scale of C 256.0 288.0 320.0 341.3 384.0 426.6 480.0 512.0 
Tempered scale 256.0 287.3 322.5 341.7 383.6 430.5 483.3 512.0 

EXERCISES 

1. Taking G as the key tone of a major scale, compute the vibra- 
tion frequency of each of the tones contained in an octave. 

2. The vibration frequency of a tone is 264. Calculate the fre- 
quencies of its third, fourth, and octave. 

3. Calculate the vibration frequency of the tone one octave below 
middle C. 

4. What is the wave length of a tone whose vibration frequency is 
256 when the temperature is 15° C. ? (See §§ 165 and 170.) 



SOUND: WAVE FREQUENCY AND WAVE FORM 185 

5. If middle C were given the frequency 260, what would be the 
frequencies oi D, A, and B ? 

6. A tone two octaves above middle C has how many vibrations 
per second ? 

7. If the keyboard of a piano extends three and a half octaves in 
each direction from middle C, calculate the vibration number of the 
lowest and the highest C on the instrument. 

8. Ascertain the interval between the pitches of two tuning forks 
making 256 and 192 vibrations per second. 

9. The tones of three bells form a major chord. One makes 200 
vibrations per second, and its pitch lies between the pitches of the 
others. Calculate the frequency of the bells. 

10. A tone is produced by a siren revolving 300 times per minute. 
What is the name of the tone produced, if the number of holes in the 
row is 24 ? 

2. RESONANCE 

185. Sympathetic Vibrations. — Vibrations that are pro- 
duced in one body by another near by which has the same 
vibration period are called sympathetic vibrations. The 
following experiments will serve to illustrate the case : 

1. Tune two wires of a sonometer, Fig. 150, so that they emit tones 
of the same pitch when plucked. Place a A-shaped paper rider astride 
one of the wires and then pluck the other. The rider will be thrown 
off, and if the wire that was plucked be stopped, a tone will be heard 
coming from the other. Throw the wires slightly out of unison and 
repeat the experiment. Only very slight vibrations will be imparted 
to the second wire. 

2. Let two mounted tuning forks having the same pitch be placed 
near together, as shown in Fig. m m 

142. Set one of the forks in vi- j| 

bration by bowing it, or by strik- ^^^^^p^^r" j ^^.. / 

ing it with a rubber stopper /d-y^E^-' /^'^S if 

attached to a wooden or glass 7 ^ f 

handle. On checking its vibra- 1^ yif 

tions a sound will be heard com- fig. 142. — Sympathetic Vibrations 
ing from the second fork. between Forks in Unison. 

Students will find it interesting to experiment in a simi- 
lar manner with a piano. For example, let the key C be 
depressed so carefully that no tone is produced. The 



186 



A HIGH SCHOOL COURSE IN PHYSICS 



damper is thus lifted from the wires of a certain vibration 
frequency. Now if the tone Q be sounded by a voice, the 
corresponding tone will be heard coming from the instru- 
ment. 

These experiments show the facility with which a body 
is put in vibration by another having the same frequency. 
The principle may be stated as follows: 

Sound waves sent out from one vibrating hody impart vi- 
brations to others^ provided the vibration frequencies are the 
same. 

186. Sympathetic Vibrations Explained. — The phenom- 
enon of sympathetic vibrations is readily accounted for 
when we consider that the body first set in vibration sends 
out periodic impulses through the air or the wood connect- 
ing the two bodies. The first impulse received by the 
second body gives it a very small amplitude of vibration. 
It returns at precisely the instant to receive the second 
impulse, and the amplitude of the second vibration is caused 

to be greater than that of the first. 
The third impulse, in like manner, 
adds energy to that already trans- 
mitted to the body. Thus, after 
the arrival of several impulses the 
amplitude of vibration of the sec- 
ond body is great enough for the 
production of an audible tone. 

187. Resonance. — A vibrating 
tuning fork when held in the hand 
produces a sound that is scarcely 
audible. Its tone may be aug- 
mented, however, by placing it 
above a properly adjusted air column that is caused to vi- 
brate sympathetically with it. 




Fig. 143. — Keen- 
forceraent of a 
Sound by an Air 
Column of Suit- 
able Length. 



SOUND: WAVE FREQUENCY AND WAVE FORM 187 



Let a vibrating tuning fork be held over a tall cylindrical jar, as 
shown in Fig. 143. By pouring water into the jar while the fork is in 
this position, a condition will be reached such that an intense sound 
may be heard proceeding from the jar. Pouring in more water de- 
stroys the effect. 

Let the experiment be repeated with a fork of a different pitch. 
If the pitch is higher, the air column will need to be shortened by 
pouring in more water ; if lower, the column must be longer. 

188. Resonance Explained. — Let a and 6, Fig. 144, rep- 
resent the two extreme positions of the prong of a vibrat- 
ing tuning fork. Just as the prong begins ^ 

a downward swing from a it starts a conden- — ^^^r^=» 
sation downward in the tube. When the 
prong begins its upward swing from 5, it 
starts a condensation in the air above it. 
Now, in order to have a reenforcement of 
the tone emitted, the condensation started 
downward in the tube must be reflected by 
the water and return in time to unite with 
the condensation produced above the prong 
as it moves upward. Hence the condensa- 
tion must travel down the tube and back while the prong 
of the fork moves from a to 5, i.e. during one half a vibra- 
tion of the fork. In a similar manner the rarefaction 
started in the tube as the prong leaves h must return from 
the bottom of the tube in time to unite with the rarefaction 
produced above the prong. 

189. Resonance Tubes and Wave Length. — In the ex- 
planation of the phenomenon of resonance given in the 
preceding section, it is clear that the waves in the air of 
the resonance tube travel twice the length of the air space 
within the tube while the fork is making one half a vibra- 
tion. The wave therefore goes four times the length 
of the air column during a complete vibration of the fork. 
But a wave progresses one wave length during a complete 



Fig. 144.— The 
Cause of Reso- 
nauce. 



188 



A HIGH SCHOOL COURSE IN PHYSICS 



vibration of a sounding body. Hence the length of the 
vibrating air column is one fourth of the length of the air 
wave produced by the fork. 

Experiments performed with tubes of different sizes 
have shown that the diameter of the tube influences the 
length of tube required for the best effect. It has been 
found that the length of the air column must be increased 
by two fifths of its diameter in order to equal a quarter of 
a wave length. 

Tuning forks are often mounted on properly adjusted 
boxes whose air spaces act as resonators when the forks 
are sounded, as shown in Fig. 142. 



c 




r 


c 




r 


c 


u>\ 


II 






i 


T 


c' r' c r' 


(2) 








1 





Fig. 



145. — Two Longitudinal Waves in the 
Condition Necessary for Interference. 



3. WAVE INTERFERENCE AND BEATS 

190o Destruction of One Wave by Another. — Let us 
imagine two trains of sound waves, (1) and (2), Fig. 145. 

Let the waves in the 
two trains be of the 
same length, i.e. pro- 
duced by sounding 
bodies of the same 
pitch. If the con- 
densations in one train at a given instant are at c, c and 
those in the other train at c\ c', it is obvious that the two 
trains cannot move through the same air simultaneously. 
For, if the compressions are equal, the condensations of 
the first train will unite with the rarefactions t\ r' of the 
second, and the result will be no change in the density of 
the air. In other words, one train of waves will destroy 
the other. Thus two sounds may combine and produce 
silence. A study of the analogous case of transverse 
waves will help to make the matter clearer. 

Let (1) and (2), Fig. 146, represent two trains of trans- 
verse waves. Let their lengths and amplitudes be equal. 



SOUND: WAVE FREQUENCY AND WAVE FORM 189 

If, now, the crests <?, c, c of one train unite with the 
troughs ^, ^, t of the other, then one train will obviously 

destroy the other, since ^^_-l^^^^ __/rrz^-^_ ^,^:r~i^ 

the crests of the first t t 

train will just fill the c ^--^^-^ 

troughs of the second ^^zizzs^-~ ^--z^^ z^ 

and vice versa. Thus Fig. 146. — The interference of Trans- 

two trains of water waves, ^^'^^ ^^''''^ 

for example, may combine and produce a level sur- 
face. 

7^e destruction of one wave train hy another similar train 
is called interference. 

191. An Example of Interference. — Let /, /, Fig. 147, 
represent the ends of the two prongs of a tuning fork. At 

^ the moment the prong^s vibrate 

\ y toward each other a condensa- 

\ / tion is produced in the region 

\ / a and rarefactions at h, h. 

h% a ^b When the prongs move out- 

~7^ "^V- ward, a rarefaction is produced 

\. at a and condensations in the re- 

\ gions 5, b. Now since a conden- 

\ sation starts from the region a 

Fig. 147. — Regions of Silence at the instant a rarefaction starts 

near a Timing Fork. ^^^^^ j^ ^^^ ^^^^ ^^^^^^ ^j^^^.^ ^^-^^ 

be places in the air around the tuning fork where the parts 
of the waves coming from a will unite with the unlike 
parts of those from 6, and continuous interference will 
result. The regions of interference near the prongs of 
a fork are shown by the dotted lines in the figure. These 
places may be found by rotating a tuning fork held near 
the ear. Positions in which the sound becomes almost 
inaudible can thus be easily located. 



190 



A HIGH SCHOOL COURSE IN PHYSICS 



Hold a vibrating tuning fork above a resonance tube tuned to re- 
enforce the tone emitted. Rotate the fork slowly, and a position will 

be found, Fig. 148, in which the sound be- 
comes practically inaudible ; for, in this 
position, the mouth of the jar receives the 
unlike parts of waves sent out from the 
opposite sides of the nearer prong of the 
fork. 




Fig. 148. — An In- 
terference Phe- 
nomenon. 



192. Alternate Interference and 
Reenf or cement. — When it is under- 
stood how two trains of sound waves 
may combine and produce an inten- 
sified sound (§ 188) or interfere and 
produce silence (§ 190) the inter- 
esting phenomenon of beats is read- 
ily explained. Beats are always produced when two tones 
that differ slightly in pitch are sounded at the same time. 
The effect is obtained as follows : 

Tune two resonance jars to reenforce two tuning forks of the same 
pitch. Load the prongs of one of the forks with pieces of tin bent in 
such a form as to cling firmly to the fork while it is in vibration. 
This will make the pitch of this fork slightly lower than that of the 
other. Now sound the two forks simultaneously and hold them over 
the resonance tubes. Fluctuations in the intensity of the sound (i.e. 
beats) will be distinguishable at a distance of several meters. Load 
the prongs of the fork more heavily, and more rapid beats will be 
observed. 

Since the forks used in the experiment differ in pitch, 
the waves sent out will differ in length, — the longer 
waves being produced by the fork with the weighted 
prongs. Since the pitches differ but a small amount, the 
wave lengths will be almost equal. Inasmuch as the 
vibrations of the weighted fork are continually falling 
behind those of the other, at certain periods the two forks 
will be producing condensations and rarefactions simul- 



SOUND: WAVE FREQUENCY AND WAVE FORM 191 

taneously ; hence reenforcement results. A little later 
the weighted fork will have fallen one half a vibration 
behind the other, and while one fork is producing a con- 
densation, the other will be sending out a rarefaction. 
Hence, at this instant, there is interference between the 
two trains of waves. 

Figure 149 shows portions of the two trains of waves 
sent out from the forks. It will be observed that at r, r 



lzzz;ui 




Fig. 149. — The Production of Beats Illustrated. 

the two trains unite to produce the greatest reenforcement 
of the sound, while at ^ unlike portions of the waves unite 
and thus destroy the sound. The resultant wave is repre- 
sented by the line AB, 

193. Law of Beats. — Let one tuning fork make 100 
vibrations per second, and another, 101. At a given 
instant imagine each fork to be sending out a condensa- 
tion. Hence, at this instant, reenforcement takes place. 
Just one half of a second later, one fork has made 50 com- 
plete vibrations, and the other 501^. Therefore, at this 
instant one fork is producing a rarefaction, and the other 
a condensation, and thus the resulting sound is weakened. 
Hence, the intensity of the sound will increase and de- 
crease once per second. This effect constitutes one beat. 
When the difference in vibration frequency is two, the 
phenomenon occurs twice per second. In every case the 
number of heats produced per second is equal to the differ- 
ence between the vibration frequencies of the two sounding 
bodies. 



192 A HIGH SCHOOL COURSE IN PHYSICS 



4. THE VIBRATION OF STRINGS 

194. The Pitch of Strings. — Many musical instruments 
employ vibrating strings or wires on account of the fact 
that such bodies emit tones of a rich quality. The wide 
range of pitch necessary is obtained by varying the lengthy 
tension^ and mass of the strings used. Thus the lengths 
of the wires of a piano, for example, vary from those that 
are about 2 inches long to others whose lengths are 4 or 
5 feet. The short wires are of a small diameter, while the 
long ones are large and massive. The violin makes use 
of four strings of different masses, and the performer ob- 
tains the necessary pitches by fingering the strings so as 
to allow the proper length to be put in vibration. 

195. Law of Length. — The laws governing the vibration 
of strings may be shown by means of an instrument called 



Fig. 150. — A Sonometer. 

the sonometer (pronounced so nom'e ter')^ Fig. 150. The 
relation between the pitch of a string and its length may 
be shown by the following experiment : 

Adjust the tensions of the two similar wires of a sonometer until 
they emit tones of the same pitch. Set up a bridge A and place a 
second bridge under the middle of one of the wires and pluck first 
one wire and then the other. It will be observed that the tones pro- 
duced are an octave apart ; i.e. one half the original length of the 
wire produces a pitch of twice (§ 180) as many vibrations per second. 
Another test may be made by placing a bridge at C just two thirds 
of the distance AB from A. By plucking both wires as before the 
interval do-sol is easily recognized. 



SOUND: WAVE FREQUENCY AND WAVE FORM 193 

This experiment may be extended to the production of 
any interval that can be recognized. In any case it will 
be found that the ratio of the length of the vibrating 
portion ^O' to the original length AB is the inverse of 
the vibration ratio given in § 180. Thus one half the 
length produces a pitch of twice as many vibrations per 
second, and two thirds of the original length emits a tone 
of three halves of the vibration frequency. Hence, the 
vibration frequencies of strings or wires are inversely pro- 
portional to their lengths. 

196. Law of Tensions. — Let the two weights used to produce 
the tensions in the wires of a sonometer be moved as far out as possible 
on the lever arms. Set the bridges so that the wires emit tones of the 
same pitch. Now move one weight back, thus shortening the lever 
arm (§ 93) until it produces only one fourth as much tension on this 
wire as before. The pitch will now be found to be an octave lower 
when the two wires are plucked. 

This experiment may be applied to other intervals ; for 
example, four ninths of the original tension will cause the 
vibration frequencies to have the ratio of 2 : 3. Hence, 
the vibration frequencies of strings or wires are directly 
proportional to the square roots of their tensions. 

197. Law of Masses. — Let a sonometer be equipped with two 
wires of the same material, lengths, and tensions, but of diif erent sizes. 
If possible, make the diameter of one twice that of the other. Since 
the masses are as the squares of the diameters, the mass of a given 
length of the larger will be four times that of the smaller. If now 
the wires are plucked, the pitch of the smaller will be an octave 
above that of the other. 

The vibration frequencies of brings or ivires are inversely 
proportional to the square roots of their masses per unit length. 

198. Vibration of a String in Parts. — The vibration of 

a stretched string or wire is more complicated than at first 

appears. When a string is bowed or plucked, the tone 
14 



194 A HIGH SCHOOL COURSE IN PHYSICS 

emitted is a compound tone produced by the string 
vibrating as a whole simultaneously with its vibration in 
parts. The readiness with which a string vibrates in parts 
may be shown by the following method : 

Attach one end of a white silk cord about 1 ni. in length to one of 
the prongs of a large tuning fork attached firmly to the table and 
whose frequency is not more than 100. Let the other end pass over 



Fig. 151. — A Cord Vibrating as a Single Segment. 

a smooth hook or pulley and support a weight, as in Fig. 151. Set 
the fork in vibration and adjust the tension and length of the cord 
until it vibrates as a single segment, as shown. Now reduce the weight 



Fig. 152. — The Vibration of a Cord in Two Parts. 

used until the cord vibrates in two parts, Fig. 152, when the fork is 
set in motion. Under suitable tensions the cord will vibrate in any 
desired number of equal parts up to six or seven. 

The tone emitted ivhen a string vibrates as a whole is called 
its fundamental. The fundamental is the lowest tone that 
a string can produce. By the pitch of a string is meant 
the pitch of its fundamental. The vibrating portions of 
the string are called loops, or segments, and a point where 
the amplitude of vibration is zero is called a node. Loops 
are often called ventral segments and antinodes. 

199. Overtones of Strings. — The character of the tone 
produced by a vibrating string is complicated by the divi- 
sion of the string into equal parts, as shown in the preced- 
ing section. Each vibrating segment of a string produces 



SOUND: WAVE FREQUENCY AND WAVE FORM 195 

a tone that is higher in pitch than the fundamental. The 
tones emitted hy the vibrating portions of any sounding body 
are called overtones, or partial tones. 

The presence of overtones emitted when a wire is 
plucked may be detected by the sympathetic vibrations 
set up in a neighboring wire. 

Let the two wires of a sonometer be tuned in unison. Place a 
bridge under the center of one wire and set A-shaped paper riders 
near the middle of each half of this wire. If the longer wire is now 
plucked near one end, the shorter wires will be thrown into vibration, 
as the riders will show. Replace the riders, and pluck the longer 
wire at the center. Only feeble vibrations are now set ap in the 
shorter wires. Again, place the bridge under one wire just one third 
of the original length from one end, and place a rider astride the 
shorter portion. Now if we pluck the longer wire near one end as 
before, the rider will be thrown off as in the former case. Replace the 
rider and pluck the longer wire one third of its length from one end. 
The rider will remain stationary. 

In this experiment we have used the shorter wire to 
show when certain overtones are present in the tone emitted 
by the longer wire. The results indicate that when a 
wire is plucked near one end, the tone produced contains 
an overtone an octave higher than the fundamental. The 
wire not only vibrates 
as a whole, but divides 

into two parts, each Fig. 153. — a Wire Emitting its Funda- 
part having a frequency mental and First Overtone. 

double that of the fundamental. This condition is illus- 
trated in Fig. 153. When the wire is plucked near the 
center, this overtone is not present, since a node would be 
required at this point. 

200. A Series of Overtones. — The second part of the 
experiment in the preceding section shows that the wire 
divides into three equal parts, each of which vibrates with 
three times the frequency of the fundamental. In a simi- 




196 A HIGH SCHOOL COURSE IN PHYSICS 

lar manner it is possible to detect the presence of still 

higher overtones, all of Avhich are produced simultaneously 

^ ^,„ ., when the strinoc is plucked or struck 

_^_5th overtone, 6 X 256 vibs. O 1 

^♦^M^ overtone; 4 xievibs.' nearonecnd. The following table 

« ^ 2d overtone, 3 X 256 vibs. - . . , ., „ 

j( ^ 1st overtone, 2 X 256 vibs. shows thc positious aud trequencics 
^^Fundamental, 256 vibe, ^f thc ovcrtoues produccd whcu a 

middle Q string- is plucked near 

Fig. 154.— Table Showing & i 

the Overtones Produced One end. 

on a Middle c String. When the vibration frequency of 

an overtone is 2, 3, 4, 5, etc., times that of the fundamen- 
tal, it is called an harmonic. The relative intensity of the 
various overtones produced by a wire depends chiefly up- 
on the manner in which it is set in motion, the point where 
it is struck or plucked, and the rigidity, density, and elas- 
ticity of the wire. 

5. QUALITY OF SOUNDS 

201. Overtones and Tone Quality. — It is a familiar fact 
that two tones of the same pitch and intensity do not neces- 
sarily sound alike. The tones produced by a violin, for 
example, are readily distinguished from those of the piano 
or flute. The tones of one violin may differ from those of 
another. This difference is one of quality. The cause of 
this difference between tones was long a matter of study 
and investigation. It was finally explained fully by the 
German physicist Helmholtz^ (1821-1894). He tells us 
that the quality of a sound depends upon the overtones 
produced hy the sounding body and their relative intensities. 

Let a wire be plucked at one end and the tone compared with that 
emitted when the wire is plucked in the middle. Again, let the wire 
be struck with a soft rubber hammer and then with something hard. 
The tones produced will be of the same pitch but of different quality. 
This last experiment may also be made with either a bell or a tuning 
fork. 

1 See portrait facing page 196. 




HERMANN VON HELMHOLTZ (1821-1894) 

A new era in the history of Sound was created in 1862 by the 
publication of Helmholtz's Lehre von den Tonempp^ndimgen, a 
work which has been translated into English under the title Sensa- 
tions of Tones. The author recognizes musical tones as periodic 
motions of the air and distinguishes the three characteristics of 
tones as intensity, pitch, and quality. He finds that tone quality 
is due to the number and relative intensity of the upper partials, or 
overtones. Helmholtz devised hollow spherical resonators of a vari- 
ety of sizes by the aid of which he was able to analyze the human 
voice and other musical tones. By means of a large series of 
tuning forks of diflPerent pitches which -were operated electrically, 
he produced tones of such a composition as to imitate the vowel 
sounds of the human voice, the tones of organ pipes, etc. 

Helmholtz received a medical education at Berlin, and from 
1855 to 1871 was professor of physiology at Bonn and Heidelberg. 
He was led to the study of physics in his endeavor to understand 
the principles involved in the eye and ear. Later he became an 
accomplished mathematician. In 1871 Helmholtz was appointed 
professor of phyics at Berlin, and in 1888 became the first director 
of the well-known Reichsanstalt (Imperial Physico-Technical Insti- 
tute) in Charlottenburg. His death occurred in 1894. 

The first contribution of Helmholtz to the science of physics is 
h's famous treatise on the Conservation of Energy published in 
1847, a publication which was of great influence in establishing 
this doctrine. He is known throughout the medical world as the 
inventor of the ophthalmoscope, an instrument used in examining 
the interior of the eye. 



SOUND: WAVE FREQUENCY AND WAVE FORM 197 

These experiments teach that the quality of the sound 
produced by a body depends upon the manner in which the 
body is set in motion. As shown in § 199, the sound given 
out by a wire is rich in overtones when plucked near one 
end; but when plucked at the center, all the overtones 
that require a node at that point are absent. When a 
tuning fork is struck with a hard substance, high ringing 
overtones are plainly heard, while the fundamental is ex- 
tremely weak. A very different quality is produced, how- 
ever, by bowing the fork or striking it against a soft pad. 

202. Stringed Instruments. — Among the most common musi- 
cal instruments which depend on the vibration of strings and wires are 
the piano, violin, guitar, mandoHn, and banjo. The piano consists of 
a series of tightly stretched wires, which in the lower octaves are very- 
massive and vary in length from three to five feet. The. shortest wires 
of the instrument are often not more than two inches in length. When 
a key is struckj the motion is transmitted through a system of levers to 
a padded hammer which strikes the wires tuned to give the correspond- 
ing pitch. The tone emitted is greatly intensified by the vibrations 
produced in the sounding-board. When the key is released, a padded 
damper falls against the vibrating wires, thus checking the motion. 
A pressure of the foot upon the right-hand pedal of the instrument 
raises all the dampers and thus leaves the vibration of every wire 
unchecked. Changes of temperature and humidity of the atmosphere 
gradually put the wires out of tune. The necessary tuning is accom- 
plished by a suitable adjustment of the tension of each wire. 

Instruments of the violin type which are played with a bow, as well 
as the guitar, mandolin, and banjo, derive their musical tones from 
properly tuned wires or strings whose lengths may be changed at will 
by the performer. The violinist by long practice learns the exact 
position at which to press the strings against the finger board to pro- 
duce the desired tones. By this method any pitch from that of the 
lowest string to the one derived from the shortest possible portion of 
the string of highest fundamental can be produced. The difficulty is 
reduced in the case of the guitar, mandolin, and banjo by the slightly 
raised frets, or bridges, placed across the finger board to indicate the 
position of the finger. These instruments owe their wide difference 
in tone quality (1) to the nature of the string, (2) to the manner in 



198 A HIGH SCHOOL COURSE IN PHYSICS 

which the strings are put in vibration, and (3) to the size, shape, and 
material of their sounding-boards. 

EXERCISES 

1. How are the different pitches produced which are necessary in 
rendering a selection on a guitar, harp, mandolin, or cello? 

2. The bridges under a stretched wire are 4 ft. apart. Where must 
a third bridge be placed to raise the pitch a major third? a minor 
third? 

Suggestion. — See § 181 for the vibration ratios for these intervals. 

3. A wire 180 cm. long produces middle C. Show by a diagram 
where a bridge would have to be placed to cause the string to emit 
each tone of the major scale of C. 

4. What is the vibration frequency of the tone three octaves above 
middle C? What length of the wire in Exer. 3 would be required 
to produce this tone? 

5. The tension of a string is 9 kg. What tension must it have in 
order to produce a tone an octave higher ? an octave lower ? a fifth 
higher ? 

6. Write the vibration frequencies of the tones one, two, and three 
octaves both above and below middle C. 

7. Write the vibration frequencies of the first four overtones of a 
G string. Name these tones. 

8. The density of steel is 7.8, and that of brass 8.7. What is the 
vibration frequency of a steel wire, if that of a brass wire of equal 
length, diameter, and tension is 280? 

Suggestion. — Let x be the vibration frequency of the steel wire 
and then write the proportion based on the law of masses, § 197. 

9. The vibration frequency of two equal wires 155 cm. long is 300. 
How many beats per second will be heard when one of the wires has 
been shortened 5 cm.? 

10. Two middle C forks were placed near together and the prongs 
of one of them weighted with bits of sealing wax ; when both forks 
were sounded, 4 beats per second were heard. Find the frequency of 
the weighted fork. 

6. VIBRATING AIR COLUMNS 

203. Organ Pipes. — The pipe organ, flute, clarinet, 
cornet, etc., employ resonant air columns for the produc- 
tion of musical tones. The pitch of the tones emitted by 



SOUND: WAVE FREQUENCY AND WAVE FORM 199 

such columns depends mainly upon the length of the 
column. The relation between the length of a column 
and its pitch is shown by the following experiments: 

1. Measure the length of various organ pipes that are available for 
experimental purposes. Compute the ratio of the length of each pipe 
to the length of pipe giving the key tone of a major scale. This ratio 
will be found to be the inverse of the vibration ratio for the corre- 
sponding tone of the scale as given in § 180. 

2. Prepare a set of glass or metal tubes about 1 centimeter in diam- 
eter of the lengths 10, 15, 20, and 30 centimeters. Leaving the tubes 
open at both ends, blow across one end of the tube 10 centimeters 
long in such a manner as to produce an audible tone. Even if the 
sound is not loud, its pitch can usually be recognized and sung by 
members of the class. Again, blow across the end of the 20-centi- 
meter tube, and compare its pitch with that of the first tube. The 
interval between the two tones will be practically an octave, the 
shorter tube having the higher pitch. Verify this relation by using 
the tubes whose lengths are 15 and 30 centimeters. In a similar 
manner ascertain the interval between the pitches of the tones emitted 
by the tubes whose lengths are 20 and 30 centimeters. This interval 
will be about a fifth. 

Organ pipes are either open or stopped. A stopped pipe 
is formed by closing one end of a tube. The experiment 
teaches that Avhen the length ratio of two open pipes is 1:2, 
the vibration ratio is 2 : 1 ; and when the former ratio is 2 : 3, 
the latter is 3:2. Hence, the vibration frequencies of open 
pipes are inversely proportional to their lengths. 

204. Open and Stopped Pipes. — Close an end of one of the tubes 
used in Experiment 2 of the preceding section, and cause it to emit its 
tone as before. Produce also the tone of the pipe when both ends are 
open, and compare the pitches. The pitch of the closed tube is an oc- 
tave lower than that of the open one. This result may be verified by 
using any of the tubes. If an organ pipe is available, blow it first 
while open and then while closed tightly at one end. The best re- 
sults are obtained by blowing it moderately. 

The pitch of a closed pipe is an octave lower than that of 
an open one of the same length. 



200 



A HIGH SCHOOL COURSE IN PHYSICS 



205. Mechanism of an Organ Pipe. — Sectional views of 
open organ pipes made of metal and wood are shown in 

Fig. 155. Air is blown 

^Q n 11 r^^®^^ from a wind chest into 

the small chamber c and 
flows in a thin stream 
through the narrow ori- 
fice i against the lip a. 
At a the air is set in vi- 
bration, and this in turn 
puts the air contained in 
the tube in vibration. 
Since the pitch of the 
pipe is determined by its 
length, it can reenforce 
only those vibrations at 
the lip <x, whose vibration 
frequency corresponds to 
its own. 




(2) 

Fig. 155. — Open Organ Pipes. (1) A 
Metal Pipe. (2) A Wooden Pipe. 






The tube 
may be re- 
garded as a 
resonator 
for reenf orc- 
ing a tone of a particular pitch. A vibrating 
tuning fork whose pitch is the same as that 
of the tube would also set the air column in 
vibration when held near the open end. 

206. Nodes in Vibrating Air Columns. — 
When an open pipe is yielding its fundamen- 
tal, or lowest tone, a node n^ (1), Fig. 156, is 
formed at the center. The arrows indicate 
that the vibratory motion of the air on opposite sides of 
a nodal point is always in opposite directions. At the 



r 



Fig. 156. — 
Nodes in 
Open and 
Stopped 
Pipes. 



SOUND: WAVE FREQUENCY AND WAVE FORM 201 

open ends of the tube aa the air is free to vibrate 
and the density of the air remains practically un- 
changed; hence these points are often called antinodes or 
loops. Since the distance from a node to an antinode is al- 
ways equal to a quarter of a wave length (§ 189), the length 
of the entire wave is four times na, or twice the length 
of the tube. 

The case of stopped pipes is different. A node is al- 
ways formed at the closed end of the tube n, (2), Fig. 156, 
and an antinode near a. The wave length is four times 
the distance an, or four times the length of the tube. It 
thus becomes clear why the fundamental of a closed pipe 
is an octave below that of the open one. 

207. Overtones Produced by Organ Pipes. — By blowing 
an organ pipe forcibly, tones of a higher pitch than the 
fundamental will be produced. 

This is also true of the tubes ( i ) i j 2 314 
used in Experiment 2, § 203. a n a n a n a 
Just as in the case of vibrating ^^^ ' ! ^ LJ_i ^ I ^ 

strings (§198), the vibrating Fig. 157. — Overtones of Open 

body, which is the air column in ^^^^' 

this instance, divides into parts according to existing 
conditions. It is by the vibration of these parts that the 
higher tones, or overtones, are produced. Figure 157 
shows the position of the nodes and antinodes when the 
pipe is sounding its first and second overtones. In (1) it is 
shown that the tube contains four quarter waves ; hence 
the wave length is equal to the length of the tube. Simi- 
larly for the second overtone (2), the division of the vi- 
brating air column forms six quarter waves. Now since 
the vibration frequency increases as the wave length de- 
creases, the first overtone has twice the frequency of the 
fundamental ; the second, three times the frequency ; and 
so on. Thus the overtones of an open pipe are the same 



202 



A HIGH SCHOOL COURSE IN PHYSICS 



as those of vibrating strings (§ 200), and are therefore 
harmonies. 

The division of the air column in stopped pipes is other- 
wise. As we have already seen, a node is always formed 
at the closed end and an antinode at the open end. Hence, 
in the production of the fundamental tone, only one quarter 
wave is contained in the pipe. But when the first over- 
tone is emitted, a node must again be formed at the closed 
end w, and an antinode at the open end. Evidently, if 
only one other node is to exist it must be at n' (1), Fig. 

158, one third of the length of 
the pipe from the open end. 
The pipe then contains three 
quarter waves. Similarly, 
when yielding its second and 
third overtones, the column 
must divide into five and seven quarter waves respectively 
in order to retain a node at the closed end and an antinode 
at the open end. This is shown for the second overtone 
in (2). Hence in a stopped organ pipe only those over- 
tones can be produced whose frequencies are odd multiples 
of the vibration frequency of the fundamental. 



(0 



(2) I ! 2 3 .' 4 5_ 

Fig. 158. — Overtones of Closed 
Pipes. 



208. Wind Instruments. — Under this title are classed all in- 
struments whose tones are emitted by air columns. The most impor- 





FiG. 159.— (1) The Flute. (2) The Clarinet. 



tant instrument of this kind is the pipe organ. The wide range of pitch 
is derived from organ pipes (see Fig. 155) of different lengths. A 
great variety in the quality of tones is secured by the use of pipes of 



SOUND: WAVE FREQUENCY AND WAVE FORM 203 




Fig. 160. — Mouthpiece of the 
ClariDet. 



different kinds, as open and closed wooden pipes, metal pipes, reed 
pipes, etc. 

The Jlute, (1), Fig. 159, corresponds to an open organ pipe whose 
length can be changed by the aid of 
openings along the side controlled by 
the fingers. The air in the instru- 
ment is set in vibration by blowing a 
current from the lips forcibly across 
the opening at a. 

The clarinet, (2), Fig. 159, is pro- 
vided with a mouthpiece, Fig. 160, which contains a reed or tongue of 
light, flexible wood, which alternately opens and closes the aperture. 
The action of the instrument resembles that of the so-called " squawker," 
made of the stem of the dandelion or by cutting a tongue on the side 
of a quill. The current of air blown by the lungs causes the reed 
to close the opening momentarily. There is thus started through 
the tube a wave, the returning reflected pulse from which forces the 
reed outward. A series of rapid puffs is in this manner maintained 
at the end of the tube. The performer secures the various pitches 
by openings in the side of the instrument, as in the case of the flute. 
The cornet, Fig. 161^ is provided with a mouthpiece of the form 

shown in Fig. 162, within 
the large opening of 
which the lips are caused 
to vibrate. The neces- 





FiG. 161 . — The Cornet. 



Fig. 162. — Mouthpiece 
of the Cornet. 



sary range of pitch is obtained by blowing overtones and controlling 
the length of the tube by the valves. (See Fig. 154.) The trombone, 
Fig. 163, is an instrument of the cornet type, in which the pitches are 




Fig. 163. — The Slide Trombone. 



204 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 164. — An Organ Reed. 



produced by sliding the portion ah to the proper positions for chang- 
ing the length of the vibrating air column. 

209. Vibrating Reeds and Diaphragms. — Reed organs, ac- 
cordions, and mouth organs are provided with reeds similar to that 
shown in Fig. 164. A current of air blown in the direction of the 

arrow suffices to set the tongue a 
in vibration. Different pitches 
are obtained by making the reed 
of suitable length and rigidity. 

A vibrating disk, or dia- 
phragm, is employed in the tele- 
phone and phonograph. The 
description of the electric telephone will be deferred until a study 
has been made of its underlying principles. The mechanical telephone 
may be made by simply connecting two metal diaphragms with a 
strong cord or wire. The vibrations set up at one end by speaking 
against the diaphragm are transmitted by longitudinal waves in the 
wire which set up corresponding vibrations at the second instrument. 
Such telephones are used for short distances only. 

The phonograph affords an interesting application of the recording 
and reproduction of sounds by the aid of a small diaphragm. In the 
process of making a " record " a smooth, or blank, cylinder of soft ma- 
terial, as wax, is placed on the rotating shaft of the instrument, and 
against its surface is carefully adjusted a sharp cutting point attached 
to the back of the diaphragm. The tones of a voice or instrument 
produce sound waves which are collected by the horn and transmitted 
to the diaphragm. When the diaphragm vibrates, its up-and-down 
motion causes the point to engrave in the surface of the rotating cyl- 
inder a spiral groove of ever- varying depth corresponding to the com- 
plex form of the soy.nd waves which fall upon it. In the reproduction 
of sound the record is mounted on the revolving shaft, and a delicate 
stylus which is attached to the back of the diaphragm is adjusted 
in the spiral groove on the surface of the cylinder. When the cylinder 
rotates, the stylus rises and falls with the little irregularities of the 
groove and thus sets the diaphragm in vibration. Since the manner 
in which the diaphragm vibrates is governed wholly by the nature of 
the engraved record, the sounds which it emits resemble very closely 
those by which the record was produced. In many instruments of 
this type, the records are engraved on disks instead of cylinders. 



SOUND: WAVE FREQUENCY AND WAVE FORM 205 

EXERCISES 

1. IVhat is the wave length of the tone produced by an open pipe 
2 ft. long? by a closed pipe of the same length? 

2. Compute the length of a middle C open pipe, the temperature 
being 20" C. 

Suggestion. — See § 170 for the relation of wave length and pitch. 

3. A whistle may be regarded as a stopped pipe. If the cavity of 
a whistle is 1 in. long, find the vibration frequency of its tone when 
the temperature of the air with which it is blown is 25° C. 

4. By blowing across the end of a tube 6 in. long, closed at one 
end, the first overtone is emitted. Find its frequency, the tempera- 
ture being 18° C. 

5. Find the vibration frequencies of the first four overtones of a 
middle C pipe, (1) when the pipe is open at both ends and (2) when it 
is stopped at one end. 

6. When a small stream of water is allowed to run into a bottle, a 
sound is heard. Does the pitch of the tone rise or fall ? Explain. 

7. If the ear is held near the mouth of a tall jar or the open end of 
a tube, a tone will be heard. Perform the experiment, and then ex- 
plain how the so-called " sound of the sea " is heard coming from 
large sea-shells. 

SUMMARY 

1. A musical sound is distinguished from a noise by the 
isochronism of the vibrations. Pitch is governed by the 
number of vibrations per second (§§ 177 and 178). 

2. A major diatonic scale is a series of eight tones whose 
vibration numbers are to each other in the relation of the 
numbers 24, 27, 30, 32, 36, 40, 45, and 48, or which bear 
the following ratios to the first, or key tone: 1, |, |, |^, |, 
f, -1/, and 2 (§§ 179 and 180). 

3. An interval is the relation between the pitches of 
two tones. It depends entirely upon the ratio of their 
vibration numbers. The common intervals are the octave^ 
sixths fifths fourth^ major thirds and minor third. The 
intervals are expressed by definite, simple ratios (§ 181). 



206 A HIGH SCHOOL COURSE IN PHYSICS 

4. A major chord consists of three tones whose vibration 
numbers are as 4 ; 5 ; 6 (§ 182). 

5. Sharps and flats are used for the purpose of supply- 
ing the necessary tones for scales other than the scale of C. 
Scales are tempered in order that an instrument with fixed 
tones may produce all scales equally well (§§ 183 and 184). 

6. Resonance depends upon the fact that a vibrating 
body will impart vibrations to another near by, whose 
natural vibration frequency is the same as its own (§§ 185 
to 189). 

7. Two trains of waves will weaken or destroy each 
other if the condensations in one train coincide with the 
rarefactions of the other. The cancellation of sound is 
complete when the amplitudes and wave lengths are 
equal (§§ 190 and 191). 

8. The coincidence of two trains of waves which differ 
slightly in length results in the alternate reenforcement 
and weakening of the resultant tone. The fluctuations of 
intensity produced in this manner are called heats. The 
number of beats per second is equal to the difference of 
the vibration numbers of the two tones (§§ 192 and 193). 

9. The vibration frequency of a stretched string or 
wire depends on its length, mass, and tension. The 
frequency is inversely proportional to the length and the 
square root of the mass per unit length and directly pro- 
portional to the square root of the tension (§§ 194 to 197). 

10. A string as a rule vibrates as a whole and at the 
same time in parts. The tone produced by the vibration 
as a whole is its fundamental, and the tones emitted by 
the vibrating parts are its overtones (§§ 198 and 199). 

11. Since a string divides into equal parts, i.e. into 
halves, thirds, quarters, etc., the vibration numbers of 
the overtones are 2, 3, 4, 5, etc., times the frequency of the 



SOUND: WAVE FREQUENCY AND WAVE FORM 207 

fundamental. Such overtones are called harmonics 
(§ 200). 

12. The quality of a sound is dependent on the over- 
tones present and their relative intensities (§ 201). 

13. The pitch of an organ pipe depends upon its length. 
The vibration numbers of the fundamentals emitted by 
pipes are inversely proportional to their lengths. The 
fundamental of a stopped pipe is an octave lower than 
that of an open pipe of equal length (§§ 203 to 205). 

14. The wave length of the fundamental of an open 
pipe is twice, and of a stopped pipe four times, the length 
of its air column (§ 206). 

15. The overtones of open pipes are all the harmonics, 
as is the case for strings ; but the overtones of stopped pipes 
are only those that have respectively 3, 5, 7, etc., times 
the vibration frequency of the fundamental (§ 207). 



CHAPTER XI 

HEAT: TEMPERATURE CHANGES AND HEAT 
MEASUREMENT 

1. TEMPERATURE AND ITS MEASUREMENT 

210. Temperature. — Among the most common experi- 
ences of everyday life are the sensations of warmth and 
coldness as we come near or in contact with objects around 
us. These sensations enable us to distinguish between 
different bodies with respect to that condition called 
temperature. If several vessels of water or pieces of iron, 
for example, are placed before us, we are able by the sense 
of feeling to arrange them in the order of their various 
temperatures. On this account, we find in common lan- 
guage many terms made use of to express what may be 
called the degree of hotness of bodies ; as hot, warm, 
tepid, lukewarm, cool, and cold. We say that one body 
is warmer than another; or, to use another expression, has 
a higher temperature than the other. 

Although our temperature sense is of great value to us 
at all times, it does not afford an infallible guide in every 
instance, as the following experiments will show : 

1. Place the hand against the wooden portion of the class-room 
seat, and then transfer it to the iron part. State which feels the 
warmer. 

2. Let three vessels of water be prepared : the first very warm, 
the second very cold, and the third lukewarm. Place the right hand 
in the cold water and the left hand in the warm water, and hold them 
there about a minute. Now transfer the left hand to the third vessel. 
The water will feel cold. Now remove the right hand from the cold 
water, and place it in the third vessel. The same water feels warm. 

208 



TEMPERATURE CHANGES AND MEASUREMENT 209 

In the first experiment the temperatures of the iron 
and wood are practically the same ; but the hand which 
is warmer than either the wood or iron falls most rapidly 
in temperature when in contact with the iron than when 
touching the wood. Furthermore, it is a well-known fact 
that a room may be considered warm by a person who has 
been running, and cold by another who has been sitting 
quietly in the house. 

211. Relation between Temperature and Heat. — When a 
body which has a certain temperature becomes hotter, we 
ascribe the cause of this change to the acquisition of heat. 
We say that heat has been added to it. When such a 
body becomes colder, we say that heat has been taken 
from it. The rise in the temperature of a body may 
be produced in many different ways, e.g. an iron is heated 
in a fire, a piece of steel by its friction with a moving 
grindstone, an electric lamp by a current of electricity, or 
the earth by the rays of the sun. 

It is necessary for us to distinguish carefully between 
the temperature of a body and its heat. The fact is well 
known that a cup of water taken from a boiling kettle is 
just as hot as the water remaining in the kettle ; or 
a pail of water dipped from a pool has at that instant 
precisely the same temperature as the pool. But it is 
also very evident that of two hot-water bottles filled 
from the same kettle, the one containing the greater 
quantity of water will give out more heat and for a 
longer time than the smaller bottle. Hence, while the 
temperatures of two bodies may be equal, the amounts 
of heat contained within them may be very different. 
Thus a thermometer placed in a vessel of water, for 
example, indicates the temperature of the water, but in 
no way does it show the amount of heat whieh the water 
contains. 

15 



210 A HIGH SCHOOL COURSE IN PHYSICS 

212. Nature of Heat. — In order to investigate the 
nature of that which brings about the changes in the tem- 
peratures of bodies, let the following experiments be per- 

_ formed : 

1. Rub the face of a coin or button against a hard- wood 
board for a minute or two. The metal will become very 
hot. 

2. Bend a piece of iron wire back and forth, and then 
feel of the place where the bending occurred. The wire 
becomes too hot to hold in the fingers. 

3. Give the end of a nail or a piece of lead a dozen 
rapid blows with a hammer. A decided rise in temperature 
will be detected. 

4. Place some tinder in the end of the piston of a "fire 
syringe," Fig. 165, and then force the piston into the cylin- 
der. When the piston is withdrawn, the tinder will be 

Fig. 165. found to be burning. 

These cases are alike in that the heat required to bring 
about the rise in temperature is produced in every instance 
at the expense of work on the part of the person perform- 
ing the experiment. Thus energy is given up hy the per- 
son^ and heat appears. Again, a moving train apparently 
loses its kinetic energy when the brakes are applied; but 
an examination of the brakes and wheels reveals the fact 
that the energy has been converted into heat, as is shown 
by a large increase in temperature. 

On the other hand, heat is constantly used in steam 
engines for hauling trains, running machinery, and for 
performing work in many other ways. 

The conclusion to which experimental results lead is 
that heat is a form of energy. The steam engine is simply 
a device for converting heat energy into a form of energy 
which can be utilized, i.e. into the energy of meclianical 
motion. When, however, this motion is checked, the 
mechanical energy is changed back into heat energy. 



TEMPERATURE CHANGES AND MEASUREMENT 211 

213. Molecular Theory Applied to Heat. — The explana- 
tion of such phenomena as we have before us is greatly 
aided by the so-called molecular theory of matter which was 
briefly stated in § 129. It is assumed that all matter is 
composed of small particles called molecules. The molecules 
of a body are separated hy small spaces within which they 
move rapidly about., probably with frequent collisions. In 
solids each molecule is restricted in its motion to a certain 
space which it does not leave. When this restriction is 
removed, the body assumes the liquid state and the only 
constraint is the mutual attraction between the molecules 
(cohesion). In gases even this last limitation is practi- 
cally removed, and hence the space occupied by a given 
mass of gas is governed only by the size of the vessel 
containing it. 

With the help of the molecular theory it is now pos- 
sible to give a more definite idea of the nature of heat. 
If m represents the mass of a molecule and v the average 
velocity of the molecules of a body, it is plain that each 
molecule possesses kinetic energy of the amount ^mv'^ 
(Eq. 2, § 62), and the entire body will have within it as 
many times this amount as there are molecules. This 
energy is called heat. Hence, we may define heat as the 
kinetic energy of molecular motion. 

214. Some Effects of Temperature Changes. — When by 
increasing or decreasing the amount of heat in a body its 
temperature is raised or lowered, one or more resulting 
changes may take place : (1) the , body may expand or 
contract ; (2) the body may undergo a change in its prop- 
erties, as in hardness, elasticity, ductility, etc. ; (3) the 
body may change its pressure against other bodies, e.g. 
a gas may increase or decrease its pressure against the 
walls of the containing vessel. 

The application of heat to a body, however, does not 



212 



A HIGH SCHOOL COURSE IN PHYSICS 



A 



always change its temperature. The body may be changed 
from a solid to a liquid, or from a liquid to a gas while 
its temperature remains constant, e.g. melting ice, boiling 
water, etc. These different classes of phe- 
nomena are commonly known as heat-effects. 

215. Measurement of Temperatures. — 
The instrument most widely used for the 
determination of temperatures is the mer- 
curial thermometer. It is constructed upon 
the principle that mercury expands when 
warmed. This thermometer consists of a 
capillary tube at the lower end of which is 
a bulb containing mercury, Fig. 166. The 
mercury completely fills the bulb and ex- 
tends some distance into the tube. Since 
the expansion of mercury is greater than 
that of glass, the thread of mercury in the 
tube rises when the temperature of the 
mercury in the instrument is increased, and 
falls when it is decreased. Before the tube 
of a thermometer is sealed, the mercury in 
the bulb is heated until it entirely fills the 
instrument, in which condition the glass 
at A is sealed off in a hot flame. When 
the mercury cools and contracts, it leaves 
a good vacuum above it in the tube. • 

216. The Fixed Points of a Thermometer. 
— In order to make it possible to compare 
the temperature measurements of one ther- 
mometer with those of another, two fixed points that are 
easily obtained are located on the tube of the instrument. 
The first of these is t\\Q freezing point of pure water. This 
point is found by packing the thermometer in ice or 
snow, as shown in Fig. 167. When the mercury has 



Fig. 166. — Tube 
and Bulb of 
a Mercurial 
Thermometer. 



TEMPERATURE CHANGES AND MEASUREMENT 213 





^)^ 



ceased to fall, the position of the end of 
the column is marked upon the tube. 

The second fixed temperature point is 
the boiling point of pure water. The ther- 
mometer is suspended over boiling water 
in a tall vessel so that the thread of mer- 
cury in the tube is completely enveloped 
by the steam, as in ¥\g, 168. The mer- 
cury rises for a time, but 
finally comes to a posi- 
tion at which it remains 
stationary. Since, how- 
ever, this temperature 
changes with the pres- 
sure of the atmosphere, 
it should be taken under 
normal atmospheric 
pressure (i.e. 760 milli- 
meters of mercury). Otlierwise a cor- 
rection must necessarily be made. 

The points thus obtained on a ther- 
mometer scale are often marked with 
Fig. 168.— Determin- the words "freezing" and "boiling," 

inff the Boiling ,^ ^ • ,^ ' , ,^ 

Point ou a Ther- thcse being the names given to the 
mometer Tube. fixed points of temperature. 
217. Graduation of Thermometer Scales. — The space 
between the freezing and boiling points is now divided 
into temperature units called degrees. According to 
the centigrade ^ scale the freezing point is marked 0°, 
and the boiling point 100°. The interval between the 
two points is then divided into 100 equal parts. Similar 
divisions are produced on the tube above the boiling 
point and below the freezing point. Centigrade ther- 

1 From centum and gradus, meaning a hundred degrees. 



Fig. 167. — Locat- 
ing the Freez- 
ing Point on a 
Thermometer 
Tube. 



214 



A HIGH SCHOOL COURSE IN PHYSICS 



mometers are almost exclusively used for scientific pur- 
poses. 

The Fahrenheit thermometer scale was introduced by a 
German physicist of that name about 1714. On this scale 



^ 



100 



2/2' 



the freezing point is marked 32°,^ and the 
boiling point 212°. The interval between 
these two points is therefore divided into 180 
equal parts, and similar divisions are laid 
off botli above the boiling point and below 
the freezing point. The Fahrenheit ther- 
mometer is the household instrument in use 
among most English-speaking people, and 
is that employed by the United States 
Weather Bureau and by physicians. 

On all thermometers, temperatures below 
the zero point are read as negative quan- 
tities. In every case the initial letter of 
the name is affixed to indicate the scale 
/rd^Mh-WlW used. For example, 25° C, 100° F. 

218. Thermometer Scales Compared. — It 
is obvious that any thermometer can be pro- 
vided with both the centigrade and the 
Fahrenheit scales, as shown in Fig-, 169. It 

Fig. 169. — Centi- .„ .„ , ' , . -, r,^ .. i 

grade and Will readily be observed that 100 centigrade 

Fahrenheit dgnrrees measure the same interval as 180 

Ihermonieter ^ 

Scales. Fahrenheit degrees. Hence, 



32' 



or 



100 centigrade degrees = 180 Fahrenheit degrees, 
1 centigrade degree = | Fahrenheit degree. 



But in order to change a temperature reading from one 
system into the other, it is necessary to take account of 
the fact that 0° F. is 32 Fahrenheit degrees below 0° C. 

1 Fahrenheit placed at the temperature which he produced by a mixture 
of ice, water, and salammoniac. 



TEMPERATURE CHANGES AND MEASUREMENT 215 



For example, 68° F. is 68 — 32, or 36 Fahrenheit degrees 
above the freezing point. But 36 Fahrenheit degrees are 
equivalent to ^ x 36, or 20 centigrade degrees. Now a 
temperature that is 20 centigrade degrees above the freez- 
ing point is 20° C. Therefore the temperature 68° F. is 
equivalent to 20° C. 

Letting F represent a Fahrenheit reading and C the 
corresponding reading on the centigrade scale, we have : 



F - 32 = f C. 



(1) 



219. Range of a Mercurial Thermometer. — The use of a 
mercurial thermometer in the measurement of high and 
low temperatures is limited by the boiling and freezing 
points of mercury. The former is about 350° C, and the 
latter — 38.8° C. The boiling of the mercury can be pre- 
vented by increasing the pressure upon it by the presence 
of nitrogen gas above it in the tube. Such thermometers 
may register temperatures up to about 500° C. For temper- 
atures below — 39° C. liquids having low freezing points 
must be used. Such a liquid is alcohol, which freezes at 
— 111° C. Many alcohol thermometers are in common use. 

220. Galileo^s Air Thermometer. — The 
first instrument designed for the meas- 
urement of temperatures was Galileo's air 
thermometer. Fig. 170. The use of this in- 
strument dates from 1593. The device con- 
sists of a vertical glass tube of small bore, 
on the upper end of which is a large bulb 
containing air. This air is warmed slightly, 
and the end of the tube is placed in some 
liquid, such as colored water. When the 
air cools, the liquid rises in the tube, being 
forced up by the atmospheric pressure. Ob- ^^^- 1"0— Gaii- 

. . leo's Air Ther- 

viously the liquid column will rise and fall mometer. 



216 A HIGH SCHOOL COURSE IN PHYSICS 

according as the temperature of the air in the bulb is re- 
duced or increased. On account of its sensitiveness to 
small changes of temperature, this thermometer is fre- 
quently used for experimental purposes. 

EXERCISES 

1. Would the range of a mercurial thermometer of given length 
be increased or decreased by reducing the size of the bulb ? by mak- 
ing the bore of the tube smaller ? Would the distance representing 
a degree be increased or decreased? 

2. If the bulb of a mercurial thermometer should permanently 
contract after its graduation, how would the fixed points be affected ? 

3. In a certain experiment only the bulb of a thermometer is ex- 
posed to the temperature that it is desired to measure. If this tem- 
perature is above that of the room, will the reading of the instrument 
be too large or too small ? 

4. How would you proceed to test experimentally the points on a 
mercurial thermometer ? 

5. The boiling point of water falls 0.1 of a centigrade degree for 
a decrease of 0.27 cm. in the atmospheric pressure. What is the 
boiling point when the barometer reads 73.3 cm.? Ans. 99° C. 

6. Reduce the following centigrade readings to the corresponding 
values on the Fahrenheit scale : 20°, 35°, 50°, -20°, -40°. 

7. How many centigrade degrees lie between the Fahrenheit and 
centigrade zero marks? 

8. The following temperature measurements were taken w^ith a 
Fahrenheit thermometer: 77°, 41°, 14°, -4°, -40°. What would a 
centigrade thermometer have indicated in each case ? 

9. The difference in temperature of two vessels of water is 25 
centigrade degrees. Express the difference in Fahrenheit units. 

10. One room is 18. Fahrenheit degrees warmer than another. 
What is the difference between their temperatures on the centigrade 
scale ? 

11. The boiling point of water at a certain place was found to be 
98.8° C. What was the atmospheric pressure at the time ? See Exer. 5. 

Ans. 72.76 cm. 
2. EXPANSION OF BODIES 

221. Linear Expansion. — We have already observed 
the use made of the expansion of mercury when heated. 



TEMPERATURE CHANGES AND MEASUREMENT 217 



It is generally true of all bodies that an increase in size 
accompanies a rise in temperature. Thus a metal rod, for 
example, undergoes an increase in length, which, although 
usually small, must be taken into account in planning 
bridges, laying railroad tracks, etc. 

1. Figure 171 ilkistrates the well- 
known ring and ball. When both 
are of the same temperature, the ball 
passes readily through the ring. When, 
however, the ball is heated, it becomes 
too large to fit the ring. If cooled, it 
will be found to resume its original size. 

2. Let a metal bar, preferably of brass, rest at one end upon 
a block of wood, as A, Fig. 172, and at the other end upon a 
round glass or metal rod B about 2 millimeters in diameter placed 




Fig. 171. — Ring and Ball. 




Fig. 172. — Expansion of a Metal Rod. 

upon a smooth block of wood. Attach a very light index of glass or 
paper about 20 centimeters long to the small rod in such a manner 
that it can move over a scale C, as shown. Now heat the bar by mov- 
ing a flame along it. Tlie movement of the index will indicate an 
elongation of the bar. When the bar cools, the index returns to the 
original position . 

222. Coefficients of Linear Expansion. — Not all solids 
expand equally. For example, a bar of copper a meter 
long expands more than a bar of iron of equal length 
for the same increase in temperature. TJie ratio of the 
increase in length of a metal bar for an increase of on*e degree 



218 



A HIGH SCHOOL COURSE IN PHYSICS 



in temperature to its length at 0° 0. is called tJie coefficient 
of linear expansion of the metal. 

Thus a rod of metal one meter long that expands 
1 millimeter when the temperature rises from 0° C. to 
100° C. has a coefhcient of linear expansion equal to 1 ^ 
(1000 X 100), or 0.00001. The coefficient of linear ex- 
pansion of a substance is expressed by the equation 



ll(.t2-ti)' 



(2) 



where l^ and l^ are the lengths before and after heating, 
and ^2 and t^ are the initial and final temperatures. 



COEFnCIENTS OF LiNEAR EXPANSION 



Aluminium . . 0.000023 

Brass 0.000018 

Copper .... 0.000017 

Glass 0.000009 

Invar 0.00000087 



Iron 0.000012 

Lead 0.000027 

Platinum .... 0.000009 

Steel 0.000011^ 




Fig. 173. — Compound 
Bar of Brass and Iron. 



223. Applications of Unequal Expansion. — By referring to 

the table of the coefficients of expansion of metals given in § 222, it 

will be seen that the common metals, brass 
and iron, expand unequally. Hence if two 
flat bars of these metals are riveted together, 
as shown in (a). Fig. 173, an increase in tem- 
perature will expand the brass more than 
the iron and cause the bar to bend, as in (h). 
The bending of a compound bar of two metals 

is employed in the dial thermometer in common 

use. The bar in this case is made circular in 

form and has one end fixed, as at A, Fig. 174. 

The other end is attached by means of a cord 

or chain at i? to a small axle C, which carries 

the pointer D. A rise in temperature causes 

the free end of the bar B to move inward and yiq 174. — DialTher- 

the pointer to register a higher temperature on mometer. 





TEMPERATURE CHANGES AND MEASUREMENT 219 

the graduated scale. For a fall in temperature the reverse movement 
takes place. 

The same principle is ingeniously applied to the balance vv^heel 
of a watch, in order to make the period 
of vibration independent of temperature 
changes. An increase in temperature weak- 
ens the hairspring which controls the vibra- 
tion of the wheel and lengthens its spokes, 
the effect in each instance being to make 
the watch lose time. In order to correct this 
tendency, the balance wheel is constructed 
as shown in Fig. 175. The outer rim of each 
of the compound bars A and £ is made of Fig. 175. — Balance Wheel 
brass, and the inner part of iron. An in- ofa vVatch. 

crease in temperature causes the portions A and B to bend toward 
the center just enough to keep the period of vibration constant. 

EXERCISES 

1. Ascertain how steel tires are tightened or "set "by a black- 
smith, and explain the various steps of the process. 

2. In what manner do engineers take account of the expansion 
and contraction of the rails when laying a railroad track ? 

3. Consult the table of linear coefficients of expansion, and ascer- 
tain why platinum wires can be sealed in glass without danger of 
breakage when the glass cools. Examine an incandescent lamp bulb, 
and see that this is the case. 

4. Why is one end of a long steel bridge often supported on rollers ? 

5. Invar is an alloy of nickel and steel. Consult the table of linear 
coefficients of expansion, and show why it is a valuable metal from 
which to make tapes for measuring, standards of length, and pendulum 
rods. 

6. Glass stoppers can often be loosened by carefully heating the 
neck of the bottle. Explain. 

7. How much does the length of a 90-foot steel rail vary if the 
extremes of temperature are — 24° C. and 35° C. ? 

8. At the temperature of 0°C. an iron pipe is 100 ft. long. What 
will be its length when steam at 100° C. is passing through it? 

9. A metal rod is 60 cm. long and expands 1.02 mm. when the 
temperature is raised from 0° C. to 100° C. Compute the coefficient 
of linear expansion of the metal. 



220 A HIGH SCHOOL COURSE IN PHYSICS 

224. Cubical Expansion. — In general, substances when 
heated expand in all directions, i.e. a rise in temperature 
is accompanied by an increase in volume. This may be 
shown to hold true for water by the following experi- 
ment: 

Fill a flask with water, and insert a rubber stopper through which 
passes a small glass tube about 40 centimeters long. Some of the 
liquid will rise in the tube. Mark the position of the top of the 
liquid column, and set the flask in a vessel of warm water. The liquid 
at first falls slightly in the tube as the flask expands and then rises 
slowly because of the increase in volume of the water. 

225. CoeflBLcients of Cubical Expansion. — The cubical 
expansion of a substance is related to volumes in the same 
manner as linear expansion is related to lengths. Thus 
the coefficient of cubical expansion of a substance is the ratio 
of the increase in volume for a change of one degree in tem- 
perature., to the volume at 0° O. 

The cubical expansion of a substance may be expressed 
by the equation 

k = X.^, (3) 

where v^ and v^ are the volumes at the temperatures t° C. 

and 0° C. respectively. This equation is easily reduced 

to the form 

Vi=Vo(l4-kt). (4) 

The coefficient of cubical expansion of any substance is 
three times the coefficient of the linear expansion of that 
substance. Hence the cubical expansions for the sub- 
stances given in the table in § 222 are easily computed. 

226. Abnormal Expansion of Water. — If a quantity of 
water at freezing temperature is warmed, its volume 
decreases until it reaches a temperature of 4° C. Upon 
being heated above this point it expands as do other 



TEMPERATURE CHANGES AND MEASUREMENT 221 

liquids. This exception to the general rule is of vast 
importance in nature. As the atmosphere grows colder, 
the water at the surface of lakes and ponds falls in tem- 
perature, and its density at first increases. The surface 
layers of water then sink and are replaced by the warmer 
water from below. This continues until the temperature 
of 4°C. is reached, when further cooling makes the sur- 
face water less dense than that underneath. Hence the 
water that has cooled below 4° C. does not sink, but 
remains at the top and is frozen. Thus, since practically 
the entire quantity of water in a lake must be reduced to 
4° C. before the surface is frozen, ice is much slower in 
forming on deep bodies of water than on shallow ones. 
Furthermore, on account of the fact that the colder ice 
and water at the top do not transmit the heat rapidly 
away from the warmer layers of the liquid below, the 
unfrozen water remains at a temperature of about 4° C. 
Hence, animal life flourishes at the bottom of a lake in 
winter, even when a little lower temperature would prove 
fatal. 

227. Cubical Expansion of Gases. — When the temper- 
ature of a gas is raised, its volume is increased, unless 
the expansion is prevented by an increase of the pres- 
sure upon the gas. Under a constant pressure the ex- 
pansion of gases is much greater than that of liquids or 
solids. When the temperature of a gas is increased, the 
change is simply an increase in the average speed of its 
molecules. As a consequence, there results an increase in 
the force delivered by each molecule in its collision with 
the sides of the containing vessel. There will also be an 
increase in the number of blows delivered per second. 
Now the pressure of the gas outward is nothing more 
than the result of this continuous bombardment of mole- 
cules ; and thus an increase in the average molecular 



222 A HIGH SCHOOL COURSE IN PHYSICS 

speed produces a corresponding increase in the pressure 
of the gas. By permitting the gas to expand, the pres- 
sure may be kept constant. Under a constant pressure, 
all gases have the same coefficient of cubical expansion (Law 
of Charles^). This number has been found experimen- 
tally to be 2-y3» or 0.00366, of the volume of the gas at 
0° C. Thus a gas whose volume at 0° C. is 100 cubic cen- 
timeters, when heated to 25° C. increases in volume 2% 
of 100 cubic centimeters. Its volume becomes, therefore, 
100 + 2^3 X 100, or 109.1 cubic centimeters. 

228. The Absolute Scale of Temperatures. — If a body of 
air, for example, at 0° C. is kept at a constant pressure 
and heated, its volume will increase ^\^ of its original 
volume for every degree that its temperature is raised. 
Thus, at 273° C. its volume will be doubled. On the other 
hand, if the original body of air is cooled below 0°, its 
volume will be diminished 2^3^ of its volume at 0° for every 
degree that its temperature is lowered. If, now, the volume 
were to continue to diminish at this rate until the tem- 
perature should reach — 273° C, mathematically it would 
become nothing. Practically, however, the air and other 
gases become liquids before reaching this temperature, and 
thus lose the properties of gases. A temperature 273 cen- 
tigrade degrees below the freezing point of water is called 
absolute zero^ and temperatures measured from this point 
as the of the scale are called absolute temperatures. It is 
clear that temperatures on the centigrade scale can be re- 
duced to the absolute by simply adding 273°. Thus 20° C. 
= 293° Ab., and - 40° 0. = 233° Ab. 

Although no one has ever succeeded in cooling a body 
to absolute zero, temperatures approaching within a very 
few degrees of this point have been attained by the evapo- 
ration of liquefied gases. The following table will serve 
1 Also called Gay-Lussac's Law. 



TEMPERATURE CHANGES AND MEASUREMENT 223 

to show some facts regarding the history of low tempera- 
ture production : 



Date 


Temperature 


Experimenter 


1714 

1778 
1823 
1877 
1877 
1898 
1908 


- 17° C 

-40° C 

- 102° C 

- 103° C 

-183°C 

-262°C. 

-269°C 


Fahrenheit 

Van Maruin 

Faraday 

Cailletet 

Pictet 

Dewar 

Onnes 



229. Laws of Gaseous Bodies. — Since the volume of a 
gas is doubled when its temperature is raised from 273° 
Ab. (0°C.) to 2 X 273, or 546° Ab. (273° C), and the 
increase in volume is uniform (§ 227), it is clear that the 
following law may be stated : 

The volume of a given mass of gas under constant pressure 
is proportional to its absolute temperature. 

Again, if the body of gas is confined in a vessel of suffi- 
cient rigidity to keep the volume constant at all tempera- 
tures, the pressure of the gas against the walls of the 
vessel will increase as the temperature rises. Further- 
more, the pressure will increase ^ys ^^ ^^^^ pressure at 
0° C. for every degree that the temperature is raised, and 
will decrease 273 of the pressure at 0° C. for every 
degree that the temperature is lowered. In other words, 
when the volume of a gas remains constant, the change in 
pressure takes place according to a law similar to that 
governing the change in volume. Hence 

The pressure of a given mass of gas whose volume remains 
constant is proportional to the absolute temperature. 



224 A HIGH SCHOOL COURSE IN PHYSICS 

Example. — At 25° C. the volume of a certain mass of gas is 400 
cm.3. Compute its volume when the temperature is lowered to 0° C. 
and the pressure kept constant. If the original pressure is 740 mm., 
compute the pressure after the decrease in temperature, assuming 
that the volume remains constant. 

Solution. — Letting x be the volume of the gas at 0° C, we have, 
by applying the law of volumes stated above, 

400 : a: : : 25 + 273 : 273 ; 
whence, x = 366.4 cm.^. 

In the second place, by applying the law of pressures, if x is the 

pressure at 0° C, 

740 : a; : : 25 + 273 : 273 ; 

whence, x = 677.9 mm. 

230. Laws of Charles and Boyle Combined. — Boyle's 
Law (§ 149) states that the product of the pressure and 
volume of a given mass of gas remains constant when 
the temperature remains the same. In practice, volume, 
pressure, and temperature may all vary. In such cases 
the following law will hold : 

The product of the pressure and volume of a given mass of 
gas is proportional to the absolute temperature. 

Example. — The volume of a gas collected in a vessel under a 
pressure of 740 mm. and at a temperature of 20° C. is 500 cm.^. Com- 
pute the volume that the gas would have at 0° C. and under a pressure 
of 760 mm. 

Solution. — Let x be the required volume. By combining the 
laws of Boyle and Charles we have 

740 X 500 : 760 X x :: 20 + 273 : 273 ; 

whence, x = 453.6 cm.^. 

EXERCISES 

1. What fractional part of its volume at 0° C. does a cubic meter 
of gas expand when warmed from that temperature to 50° C, the 
pressure remaining constant? What is the final volume of the gas? 



TEMPERATURE CHANGES AND MEASUREMENT 225 

2. The volume of a certain gas at 20° C. is 300 cm. 3. What is its 
volume when the temperature is reduced to 0° C, the pressure being 
constant ? 

3. The pressure exerted by a gas confined in a reservoir is 500 g. 
per square centimeter when the temperature is 10° C. What is its 
pressure when the temperature is raised to 40° C. ? 

4. The volume of a gas collected in a chemical experiment is 
30 cm.^, its temperature 25°, and its pressure 750 mm. Find the vol- 
ume of the same gas at 0° C. and under a pressure of 760 mm. 

5. If the mass of a cubic centimeter of air at 0° C. and under a 
pressure of 760 mm. is 0.001293, what will be its density in a room 
where the temperature is 22° C. and the barometer reads 745 mm. ? 

6. The quantity of air in a room 8 x 12 x 15 ft. will contract to 
what volume when its temperature falls from 20° C. to 0° C? 

7. To what temperature would the air confined in a flask at atmos- 
pheric pressure and a temperature of 10° C. have to be heated in order 
to exert a pressure of 3.5 atmospheres? 

8. Upon heating 300 cm.^ of a gas from 0° C. to 30° C, the volume 
was found to be 333 cm.^. Ascertain the coefficient of expansion of 
the gas. 

9. The capacity of a steel gas cylinder is 3 cu. ft. Illuminating 
gas is compressed in the cylinder until the pressure is 15 atmospheres 
at a temperature of 10° C. What volume will this quantity of gas 
assume when allowed to escape into a space where the pressure is 
1 atmosphere and the temperature 25° C? 

3. CALORIMETRY, OR THE MEASUREMENT OF HEAT 

231. The Unit of Heat. — ^Tlie unit employed in the 
measurement of heat is called the calorie. The calorie is 
the (quantity of heat required to raise the temperature of a 
gram of water one centigrade degree. When the tempera- 
ture of a gram of water is raised from 0° C. to 100° C, 
100 calories of heat are required. Again, to change the 
temperature of 1 kilogram of water from 20° C. to 45° C. 
requires that 1000 x (45 — 20), or 25,000 calories of heat 
be taken on by the water. Thus, if a mass of water be 
changed in temperature a given amount, the quantity of 

heat involved is measured hy the product of the mass of water 
16 



226 A HIGH SCHOOL COURSE IN PHYSICS 

and its change of temperature. Hence, to change the 
temperature of a mass of m grams of water t centigrade 
degrees, we may write for the required quantity of heat, 

H (in calories) = 

m (in grams) x t (in centigrade degrees). (5) 

232. Specific Heat. — If a given quantity of heat be ap- 
plied to equal masses of different kinds of matter, as lead, 
mercury, water, iron, and copper, the temperatures will not 
all be changed equally. The amount of heat that will 
warm a gram of water one degree (^.g. one calorie) will 
raise the temperature of an equal mass of lead or of 
mercury about 30 degrees and that of the iron or the 
copper about 10 degrees. 

Place 100 grams each of lead shot, iron cuttings, and bits of alumin- 
ium wire in three large test-tubes. Set the tubes upright in a vessel 
of boiling water, and allow them to remain there several minutes 
while the water continues to boil. Also place 100 grams of water at 
the temperature of the room in each of three beakers. Now pour the 
lead shot whose temperature is 100'' C. into the water in one of the 
beakers, stir the mixture thoroughly, and ascertain the rise in temper- 
ature of the water. Do the same with the other metals. Although 
the metals fall in temperature almost equal amounts, they deliver to 
the water very unequal quantities of heat. The aluminium will warm 
the water through about twice as many degrees as the iron and about 
six times as many as the lead. 

Experiments like those just described lead to the con- 
clusion that each gram of lead gives out, upon cooling one 
degree, about one sixth as much heat as one gram of 
aluminium. Also that a gram of iron delivers only one 
half as much heat as an equal mass of aluminium when 
cooled an equal amount. For this reason substances are 
said to differ in thermal capacity^ or specific heat. 

The specific heat of a substance is the ratio of the quantity 
of heg,t required to raise the temperature of a certaiyi mass 



TEMPERATURE CHANGES AND MEASUREMENT 227 

of it one degree to the quantity of heat required to raise the 
temperature of an equal mass of ivater one degree. 

The specific lieat of a substance is numerically equal to 
the number of calories received by one gram of the sub- 
stance when its temperature rises 1° C. Thus the heat 
required to raise the temperature of 1 g. of iron 1° C. is 
0.113 calories; of 1 g. of lead, 0.032 calories, etc. 

233. Specific Heat Determined. — When a hot substance, 
as heated mercury, for example, is placed in cold water, 
the two bodies assume the same temperature. The heat 
given up by the substance which cools is utilized in raising 
the temperature of the water. In other words, the quantity 
of heat gained hy the cold body equals that lost by the tvarm 
body. 

Place 300 grams of lead shot in a large test-tube, and suspend the 
tube in boiling water for at least ten minutes. While the lead is 
heating, the tube should be kept closed with a cork. Place 100 grams 
of water in a beaker, and cool it a few degrees below the temperature 
of the room. Now pom- the shot quickly into the water and stir care- 
fully until the temperature of the mixture becomes stationary. Note 
and record the rise in temperature of the water. In order to complete 
the solution of the problem, we must form an equation that expresses 
the equality between the heat lost by the lead and that gained by the 
water. The following illustration will make the process clear: 

In an experiment the initial temperature of the water was 17° C, 
the temperature of the mixture 24.2° C, and the boiling point 100° C. 

Let X be the specific heat of lead. 

The fall in temperature of the lead = 100" - 24.2°. 

The heat lost by the lead = (100 - 24.2) 300 x calories. 

The rise in temperature of the water = 24.2° — 17°. 

The heat gained by the water = (24.2 — 17) 100 calories. 

Hence, (100 - 24.2) 300 x = (24.2 - 17) 100 ; 

whence, x = 0.0316. 

This process for determining the specific heat of a sub- 
stance is known as the '^ method of mixtures." It can be 



228 A HIGH SCHOOL COURSE IN PHYSICS 

applied successfully to a large number of substances that 

do not dissolve when placed in water. The specific heats 

of some common substances are given in the following 

table : 

Specific Heats 

Aluminium .... 0.212 Mercury 0.033 

Copper 0.093 Platinum ...... 0.032 

Ice 0.502 Silver 0.056 

Iron ....... 0.113 Tin . 0.055 

Lead 0.032 Zinc 0.093 

EXERCISES 

1. A mass of 75 g. of water is cooled from 95° C. to 32° C. How 
much heat is given up ? 

2. A 100-gram mass of copper rises in temperature from 15° C. to 
100° C. How much heat does it absorb? 

3. If 500 calories are applied to 500 g. of mercury at 10° C, to 
what point will the temperature of the mercury rise? 

Suggestion. — Let x be the final temperature of the mercury. 

4. A mass of iron weighing 400 g. and having a temperature of 
98° C. is placed in 100 g. of water at 14° C. ; the temperature of the 
combined masses is 40° C. Compute the specific heat of the metal. 

Suggestion. — See example given in § 233. 

5. Find the resulting temperature when 400 g. of water at 90° C. 
are mixed with 150 g. of water at 10° C. 

Suggestion. — If a: is the resulting temperature, the former mass 
falls in temperature 90 — a; degrees, while the latter rises a: — 10 
degrees. Express the equality between the heat lost by the former 
and that gained by the latter. 

6. A vessel contains 250 g. of lead shot and 150 g. of water. Find 
the amount of heat required to raise the temperature of the mixture 
from 15° C. to 80° C. Ans. 10,270 calories. 

7. The temperature of a block of ice weighing 100 kg. rises from 
— 15° C. to the melting point. What quantity of heat is absorbed ? 

8. If 100 g. of aluminium at 97° C. were dropped into 50 g. of 
water at 10° C, what would be the temperature of the mixture? 



TEMPERATURE CHANGES AND MEASUREMENT 220 

SUMMARY 

1. Temperature is the degree of hotness of a body and 
is the condition which determines in what direction a 
transfer of heat will take place (§§ 210 and 211). 

2. Heat is a form of energy into which all other forms 
are convertible. It is the kinetic energy of the moving 
molecules of a body (§§ 212 and 213). 

3. Changes in the temperature of bodies produce 
(1) expansion and contraction, (2) change in their prop- 
erties, and (3) changes in pressure. The application of 
heat may change the state of matter without producing a 
change in temperature (§ 214). 

4. The mercury thermometer makes use of the expan- 
sion of mercury in the measurement of temperatures. 
Each instrument is graduated according to the position of 
two fixed points^ the boiling point and the freezing point 
of pure water (§§ 215 and 216). 

5. On the centigrade scale the freezing point is marked 
0°, and the boiling point (under a pressure of 760 mm.) 
is marked 100°. On the Fahrenheit scale the correspond- 
ing points are marked 32° and 212°. The relation be- 
tween temperatures expressed on the two scales is 

i^- 32 = f (7 (§§ 217 and 218). 

6. The coefficient of linear expansion of a body is the 
ratio of its increase in length for an increase of 1° C. to 
its length at 0° C. (§§ 221 to 223). 

7. The coefficient of cubical expansion of a substance is 
the ratio of its increase in volume for a change of 1° C. 
to its volume at 0° C. (§ 225). 

8. The volume of a given mass of water when heated 
from 0° C. contracts until the temperature reaches 4° C. 
Further heating causes it to expand. Hence the greatest 
density of water is at 4° C. (§ 226). 



230 A HIGH SCHOOL COURSE IN PHYSICS 

9. The coefficient of cubical expansion of all gases is 
practically the same and is expressed by the fraction 273 
(§ 227). 

10. Absolute temperatures are measured from absolute 
zero, which is the same as —273° C. Hence temperatures 
measured on the centigrade scale are reduced to the abso- 
lute by adding 273° (§ 228). 

11. The volume of a given mass of gas under constant 
pressure is proportional to its absolute temperature 
(§ 229). 

12. The pressure of a gas under constant volume is 
proportional to its absolute temperature (§ 229). 

13. The product of the pressure and volume of a given 
mass of gas is proportional to its absolute temperature 
■(§ 230). 

14. Heat is expressed in terms of a unit called the 
calorie. The calorie is the quantity of heat required to 
raise the temperature of 1 g. of water 1° C. (§ 231). 

15. Like masses of different substances require different 
quantities of heat to produce equal changes of temper- 
ature, i.e. they differ in specific heat. The specific heat of 
a substance is the ratio of the quantity of heat required 
to raise the temperature of a certain mass of it 1° C. to 
the quantity required to raise the temperature of an 
equal mass of water the same amount (§ 232). 

16. The quantity of heat required to raise the tem- 
perature of any mass of a substance a given amount is the 
product of the mass, the rise in temperature, and the 
specific heat of the substance (§ 233). 

17. Specific heat is determined by the "method of 
mixtures" (§233). 



CHAPTER XII 

HEAT: TRANSFERENCE AND TRANSFORMATION OF 

HEAT ENERGY 

1. CHANGE OF THE MOLECULAR STATE OF MATTER 

234. Fusion. — If the motion of the molecules of a body 
is principally of a vibratory nature, we find difficulty in 
changing the form of the body and, therefore, call it a 
solid. When, however, by the application of heat, the 
molecular motion is so increased that the particles break 
away from the constraining forces and thus overcome 
their "fixedness of position," the mass assumes the prop- 
erty of fluidity and becomes a liquid. Hence the state in 
which a substance exists depends largely upon its tem- 
perature. Thus mercury, which is a liquid under ordinary 
conditions, changes into a gas at 350° C. and remains a 
solid at all temperatures below — 39° C. The temper- 
ature at whfch a solid changes into a liquid is called its 
melting or fusing point, and the process is called fusion. 
The reverse change, i.e. from a liquid to a solid, is called 
solidification. In the case of water, the process is called 
freezing. 

Collect a quantity of snow in a vessel, preferably when the out-door 
temperature is several degrees below the freezing point. In summer 
broken ice must be used. Place a thermometer in the snow so that it 
can be read easily and set the vessel outside the window. When ready 
to proceed with the experiment, bring the snow into the warm room, 
and read the thermometer at brief intervals. The temperature of the 
snow will be found to rise gradually to 0° C, where the mercury re- 
mains while the snow is melting. The vessel of snow may be gently 
heated, and, if the mixture is kept thoroughly stirred, the temperature 

231 



232 A HIGH SCHOOL COURSE IN PHYSICS 

will remain 0° C. Continued heating after the snow has melted will 
raise the temperature of the resulting liquid. 

The melting point of a crystalline substance, as snow, is 
well marked, but amorphous bodies, as glass, tar, pitch, 
etc., pass through a semiliquid state several degrees below 
the temperature of liquefaction. It is due to this property 
that glass can be formed into vessels of any desired shape, 
and that wrought iron can be fashioned according to the 
demands of the blacksmith. 

235. Laws of Fusion. — The laws governing the fusion 
of crystalline substances and the reverse change of solidifi- 
cation are as follows : 

(1) A crystalline substance under a constant pressure has 
a definite fusing point which is also the temperature at which 
solidification takes place. 

(2) When a crystalline substance begins to melt., its tem- 
perature remains constant until all of it is liquefied. 

(3) A substance that contracts while melting has its fusing 
point slightly lowered by increased pressure, but a substance 
that expands while melting has its fusing point slightly raised 
by an increase of pressure. 

(4) In the presence of a dissolved substance, the solid 
forms by crystallization at a temperature below the freezing 
point of the pure solvent. 



Mercury . 
Ice . . . 

Benzine . 
Acetic acid 
Paraffin . 



Melting Points 

39.° C. Tin ..... . 232° C. 

0.°C. Lead 325° C. 

7.°C. Silver 954° C. 

16.° C. Copper 1100° C. 

55.° C. Cast iron .... 1200° C. 



236. Effect of Pressure on the Fusing Point of Ice. — It 

is well known that water expands when it freezes. This 



HEAT: TRANSFERENCE AND TRANSFORMATION 233 

is shown by the floating of ice and the bursting of frozen 
water pipes. Ice therefore contracts when it melts. 
Hence, according to the third law of fusion stated in 
§ 235, an increase of pressure lowers the melting or fusing 
point. The lowering is very slight, amounting to about 
0.0075° C. for an increase of one atmosphere. 

1. Let two pieces of ice be pressed firmly r-^-^^^ r "z z::i _i.a„^ 
together. When the pressure is removed, the L' :/ } *4'*A 
pieces will be found to be frozen together. j^-^-^^-^^ia . hqiilr^L^^^^ 

2. Connect two heavy weights by means of " 
a strong wire, an& hang them over a block of 
ice, as shown in Fig. 176. In a short time the 
wire will cut into the block and, at last, en- 
tirely through, leaving the ice still in one piece. ^^^- ^'^^- "~ ^^^'^ Cutting 

through a Block of 

In each of these cases the melting ice. 
point at the places where the pressure is applied is slightly 
lowered; hence some of the ice. melts, forming a film of 
water a little below 0° C. When the pressure is removed 
the film, which is at a temperature below the freezing point, 
solidifies, cementing the two pieces of ice together. In Ex- 
periment 2 the melting under pressure takes place below 
the wire ; then, as the liquid flows up above the wire, the 
pressure is removed, and it freezes. This process is 
known as regelation (pronounced rege Id'tion). 

237. Heat of Fusion. — In general, the fusion of any 
solid requires the application of heat. If the substance 
is of a crystalline structure, as ice, the heat energy im- 
parted to it does not sensibly raise its temperature during 
the melting process. In all such cases the energy sup- 
plied to the solid is used in producing the change of state. 
Non-crystalline solids, such as waxes, iron, glass, etc., 
become plastic when heated, and have no definite melting 
point. 

1. Note the temperature of snow or finely chipped ice placed in a 
metal vessel. Place a flame under the vessel, and allow some of the 



234 A HIGH SCHOOL COURSE IN PHYSICS 

ice to melt. Remove the flame, stir the contents of the vessel well, 
and again take the temperature. Apply more heat, and note the 
temperature after stirring. In every case the temperature will be 
found to be the same. 

2. Place equal quantities of ice and ice water in two similar vessels, 
and set both in a large vessel of hot water placed over a flame. In one 
vessel heat is changing ice into water; in the other the temperature 
of the water is being raised. If the contents of the vessels are con- 
tinually stirred, it will be found that when the ice is melted, the tem- 
perature of the water will have risen to about 80° C. 

Heat energy is applied about equally to the ice and ice 
water in Experiment 2. That applied to the cold water 
increases the average kinetic energy of the molecules, 
(§ 213), and thus the temperature is raised. The heat 
applied to the ice, however, does not produce any increase 
in the kinetic energy, but suffers a transformation into 
potential energy by producing molecular separation in op- 
position to the mutual attraction (cohesion) between the 
particles that compose the body. In other words, the heat 
energy expended in meltiyig the ice has ceased to he heat and 
simply represents the work necessary to change the ice 
from the solid to the liquid state. The number of calories 
per gram required to liquefy a substance without producing 
any change in its temperature is called the heat of fusion^ 
of that substance. 

238. Heat of Fusion of Ice Measured. — The quantity of 
heat required to melt a gram of ice can be measured by 
the method of mixtures as shown in the following experi- 
ment: 

Let 300 grams of water at about 35° C. be placed in a beaker and 
its temperature accurately noted. Prepare also a quantity of ice in 
lumps about as large as walnuts. Dry the pieces of ice with a towel, 
and drop them in small quantities into the warm water. Stir 
thoroughly to melt the ice. When enough ice has been added and 

1 Sometiines called " latent " hpat of fusion. 



HEAT: TRANSFERENCE AND TRANSFORMATION 235 

melted to make the temperature of the water about 5° C, weigh the 
contents of the beaker and compute the mass of ice melted. An equa- 
tion is now formed between the heat lost by the water and that gained 
by the ice. An illustration will make the process clear. 

Tn an experiment the temperature of the warm water was 36.5^ C. 
After adding 110 grams of ice the resulting temperature of the water 
was 5.5° C. 

Let X be the heat of fusion of ice. 

The heat lost by the warm water = 300 (36.5 — 5.5) calories. 

The heat required to warm 110 

grams of water formed by the 

melted ice from O'' C. to 5.5° C. = 110 x 5.5 calories. 
The heat required to melt the ice = 110 x calories. 
Hence, 110 2; + 110 x 5.5 = 300 (36.5 - 5.5) ; 

whence, x = 79.9 calories. 

Careful investigation has shown that the heat of fusion 
of ice is 80 calories. 

239. Heat Given out by Freezing Water. — According 
to the doctrine of the Conservation of Energy (§ 64), 
heat energy equivalent to that which is required to 
change a solid into a liquid must be given out when the 
reverse change (i.e. solidification) takes place. 

Make a freezing mixture of snow and salt stirred well together. 
Into this mixture, which will be several degrees below 0° C, set a 
test-tube containing water and a thermometer. If the water in the 
tube is not disturbed, it may reach a temperature below 0"^ C. without 
freezing. If, however, the thermometer be moved gently against the 
wall of the test-tube, the water quickly begins to freeze and the tem- 
perature rises at once to 0° C. 

In the freezing mixture of snow and salt used in the 
experiment, both solids pass into the liquid state. Just as 
in the case of fusion (§ 238) heat is absorbed by both the 
salt and snow during the change. The heat energy 
(kinetic energy) acquired by the solids in liquefying is 
converted into potential energy in giving their molecules 
the molecular freedom which exists in a liquid. In fact 



236 



A HIGH SCHOOL COURSE IN PHYSICS 



the solids cannot liquefy unless they can acquire heat 
somewhere. In this case the heat is taken from the water 
in the test-tube, and thus its temperature is lowered and 
a portion solidified. On the other hand the rise in tem- 
perature indicated by the thermometer which was placed 
in the test-tube shows that heat is given out when the 
water changes to ice. 

When a gram of water freezes^ 80 calories are given out to 
its surroundings. The enormous amount of heat evolved 
by the freezing of water in large lakes is of great eco- 
nomic importance, since it prevents large and sudden falls 
of temperature in the vicinity. 

EXERCISES 

1. How does the presence of tubs of water in a cellar tend to pre- 
vent the freezing of vegetables? 

2. What has the large heat of fusion of ice to do with the rapid- 
ity with which snow and ice disappear on a warm day? 

3. In freezing cream a metal vessel B (Fig. 177) containing it is sur- 
rounded by a rapidly liquefying mixture of 
ice and salt, A. Give a complete explanation 
of the process. 

4. Can a piece of ice be warmed above 
0° C. ? Can it be cooled below 0° C. ? 

5. Find the amount of heat required to 
melt 50 g. of ice at 0° C. and to raise the tem- 
perature of the resulting water to 15° C. 

6. A kilogram of ice at 0° C. is placed in 
Fig. 177. — Liquid B Fro- an equal mass of water at 100° C. Find the 

zen by the Melting of resulting temperature. ' Ans.lO°C. 

Ice in A. wj ^ piece of ice weighing 100 g. and 

having a temperature of — 15° C. is brought into a room where the 
temperature is 30° C. What thermal processes take place ? What 
quantity of heat is involved in each process ? 

8. What mass of ice at 0°C. will be required to reduce the tem- 
perature of a kilogram of water from 100° C. to 20° C. ? 

Suggestion. — Let x be the required mass, and form an equation 
similar to that used in § 238. 




HEAT: TRANSFERENCE AND TRANSFORMATION 237 

240. Evaporation and Ebullition. — We have seen in 
§ 235 that the change from the solid to the liquid state 
takes place at a definite temperature in crystalline sub- 
stances. The change, however, from the liquid to the 
gaseous state occurs at all temperatures by the slow pro- 
cess of evaporation. The gas that rises from a liquid sub- 
stance is called the vapor of that substance. Even ice and 
ice water evaporate. But by sufficient heating a liquid 
reaches a certain temperature at which the familiar pro- 
cess of boiling, or ebullition, begins. This temperature, 
which is called the boiling pointy varies greatly with dif- 
ferent substances and with the atmospheric pressure. 

241. Evaporation Explained. — The process of evapora- 
tion is made clear by the help of the molecular theory 
(§ 213). At the exposed surface of a liquid the velocity 
of many of the molecules is sufficient to enable them to 
break through the surface beyond the range of attraction 
of the molecules of the liquid. If the temperature be 
raised, the average velocity of the molecules in the liquid 
is increased, and a more rapid surface loss will result. 
The removal of a large number of the most rapidly mov- 
ing molecules in this manner decreases the average kinetic 
energy of those that are left behind. Consequently, the 
temperature of the remaining liquid is lowered by the pro- 
cess. Evaporation always takes place at the expense of the 
heat energy contained in the liquid. 

242. Laws of Evaporation. — (1) The rate of evaporation 
becomes greater as the exposed or free surface of the liquid 
is iyicr eased. 

A pint of water, for example, will evaporate faster when 
placed in a broad, shallow pan than when left standing in 
a pitcher. When spread over the floor, it disappears in a 
short time. Hence a wet cloth will dry faster when spread 
out than when left folded. 



238 A HIGH SCHOOL COURSE IN PHYSICS 

(2) The rate of evaporation becomes greater as the tempera- 
ture of the liquid and vapor is increased. 

As the temperature rises, the increased molecular motion 
enables molecules to break away from the surface at a 
greater rate. 

(3) The rate of evaporation is increased hy the removal of 
the vapor from the space above the liquid. 

When the space above the liquid contains a quantity of 
the vapor, a great number of the vapor molecules moving 
about in all directions by chance strike the surface and 
reenter the liquid. The greater the number of molecules 
present in the vapor, the larger will be the number which 
return to the liquid. The return of the molecules which 
have once detached themselves from the liquid can be pre- 
vented by removing the vapor as fast as it is formed. 
This accounts for the fact that roads dry quickly on windy 
days, and ink is frequently evaporated by blowing upon 
the paper. 

243. Vapor Pressure. — When a liquid is placed in a 
vacuum, it rapidly evaporates until a condition is reached 
when the quantity of vapor present becomes constant, 
i.e, when the number of molecules leaving the liquid per 
second is just equaled by the number which reenter it in 
the same length of time. The vapor is then said to be 
saturated. A vapor, like any other gas, exerts a pressure 
against the walls of the containing vessel. The amount 
of pressure exerted depends upon the temperature ; if 
the temperature rises, more of the liquid evaporates and 
the pressure increases; if the temperature falls, some of the 
vapor condenses and the pressure decreases. The pressure 
exerted by a saturated vapor above its liquid is called the 
maximum vapor pressure at the existing temperature. For 
example, if a vessel of water be placed in a vacuum, it 



HEAT: TRANSFERENCE AND TRANSFORMATION 239 

vaporizes at 0°C. until the vapor pressure is 4.6 millime- 
ters of mercury; at 10° C, 9.16 millimeters; at 20° C, 
17.39 millimeters, etc. 

It is a peculiar fact that the maximum pressure exerted 
by a particular vapor in a closed space is independent of 
the pressure of other vapors that may be present. In other 
words, the quantity of vapor required to produce saturation 
in a given space is the same whether that space is a vacuum 
at the beginning or is occupied by other vapors. 

244. Unsaturated Vapors. — When the vapor present in 
a given space is not enough to produce the condition of 
saturation, i.e. to produce the maximum vapor pressure at 
that temperature, the vapor is called unsaturated vapor. 
This will always be the case in a closed space in which an 
insufficient quantity of liquid is placed. On the other 
hand, when a vapor is kept in contact with its liquid, it is 
always saturated. For example, the vapor above a liquid 
in a tightly corked bottle is saturated, and evaporation 
cannot occur; but when the bottle is open, the vapor 
is always slightly unsaturated, and therefore a continual 
change of the liquid into a vapor takes place. 

245. Atmospheric Humidity. — Since evaporation of water 
is always taking place at the surface of lakes, rivers, and 
other bodies of water, and also from the soil and vegetation, 
there is always more or less water vapor present in the 
atmosphere. That this is the case may be shown by the 
following experiment: 

Fill a polished vessel or a glass beaker with ice water, and allow it 
to stand exposed to the air. In a short time drops of moisture will be 
seen forming on the exterior surface of the vessel. 

Ordinarily the air does not contain saturated water 
vapor. But since the quantity of vapor necessary to pro- 
duce the condition of saturation in a given space is less at 
low temperatures, the air in contact with the cold vessel 



240 A HIGH SCHOOL COURSE IN PHYSICS 

soon reaches a temperature at which the water vapor 
already present becomes saturated. When the tempera- 
ture is reduced below the point at which the water vapor 
is saturated, the vapor is condensed, and moisture is de- 
posited upon the cold surface of the vessel. The tempera- 
ture at which moisture begins to form from the atmospheric 
water vapor is called the dew-point. 

We think of the air as being dry or moist (i.e. arid or 
humid^ according as we feel that it contains little or much 
water vapor. These conditions of the air, however, involve 
(1) the amount of vapor actually present and (2) the quantity 
necessary to produce saturation under the given conditions. 
It is upon the relation of these two elements that the sen- 
sations of dryness and moisture depend. The condition of 
the air^ in regard to the water vapor which it contains, is 
expressed by the ratio of the mass of water vapor in a given 
volume of air to the mass of vapor required to produce the 
condition of saturation at the same temperature. This ratio 
is called the relative humidity of the air. For example, if 
the quantity of water vapor actually present in a given 
space is 15 grams, and the amount required to produce 
saturation at that temperature is 20 grams, the relative 
humidity is f , or 75 % . 

If air containing water vapor is caused to undergo a 
decrease of temperature, the relative humidity increases 
since the cool air is nearer to its point of saturation. If 
the cooling is carried far enough, moisture which we call 
dew is deposited on solid objects. The experiment of the 
beaker of ice water described above is an illustration of 
tliis effect. If the temperature at which the moisture is 
deposited is below 0° C, it is frozen as fast as it is formed 
and is called frost. Similarly, when any region of the air 
cools below the dew-point, particles of water produced by 
slow condensation collect about dust particles and produce 



HEAT: TRANSFERENCE AND TRANSFORMATION 241 

fogs and mists. When a condensation takes place at high 
altitudes, clouds are formed. The slowly falling cloud 
particles may unite to produce a drop of rain. In cool 
seasons condensation may take place at temperatures 
below the freezing point. In this case the result is snow^ 
sleety or hail. 

246. Ebullition. — It has already (§ 243) been stated 
that the saturation pressure (maximum vapor pressure) of 
water increases with the temperature. While at 10° C. it 
is only 9.16 millimeters of mercury, at 90° C. it is 525 milli- 
meters, and at 100° C, 760 millimeters. It is clear, there- 
fore, that at some definite temperature (viz. 100° C. for 
water) the maximum vapor pressure must be equal to a 
pressure of one atmosphere, or 760 millimeters. At this 
temperature the average speed of the molecules of the 
liquid becomes so great as to render the cohesive force be- 
tween them unable longer to reta,in them. Hence small 
groups of molecules nearest the heated areas assume 
greatly enlarged volumes (bubbles) within which practi- 
cally no cohesion exists, because of the vastly increased 
distance between the particles. Thus at this temperature 
ebullition, or boiling, takes place. This phenomenon is 
marked by the formation of bubbles of saturated vapor 
that rise to the surface and burst. The temperature at 
which this condition of the liquid is reached is the boil- 
ing point of the liquid. 

247. Laws of Ebullition. — l. Fit a 2-hole rubber stopper in a 
test-tube. Thrust a thermometer through one of the holes and an open 
glass tube through the other. Place a small quantity of sulphuric 
ether in the test-tube, and hold it in a vessel of water at a temperature 
of about 70° C. Soon the ether will begin to boil, and the thermome- 
ter will indicate a steady temperature of about 35° C. (Caution. — 
On account of the high inflammability of ether vapor, the tube con- 
taining it should not be brought near a flame.) Place a finger over 
the end of the open glass tube, and thus carefully allow the pressure 

. 17 



242 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 178. — Water Boiling 
under Reduced Pressure. 



of the ether vapor to increase. The boiling point will be observed to 

rise several degrees. 

2. Let a round-bottomed flask be half filled with water, and the 

water boiled for two or three minutes to 
enable the steam to expel the air. Close 
the flask with a rubber stopper, and in- 
vert it on a stand, as shown in Fig. 178. 
Although the temperature of the water 
will fall rapidly, the water can be made 
to boil vigorously by pouring cold water 
upon the flask. 

3. Set a beaker of water at 90° C. or 
less under the receiver of an air pump 
and begin to exhaust the air. The 
water will boil vigorously as long as the 
pressure is kept sufficiently reduced. 

The experiments just described illustrate the following 
general laws of ebullition : 

(1) Every liquid has its otvn boiling pointy which is in- 
variable under the same conditions. 

(2) The boiling point of a liquid rises or falls as the 
pressure upon the liquid increases or decreases. 

The cold water poured upon the flask containing steam 
causes a portion of the vapor to condense. This reduces 
the pressure within the flask, thus lowering the boiling 
point to the temperature of the water. If the air has been 
very thoroughly expelled from the flask, the water may be 
kept boiling until it is scarcel}^ lukewarm. 

Because of the decrease of atmospheric pressure with 
an increase of altitude, the boiling point of water at 
Altman, Colo., probably the highest town in the United 
States, is about 88.5° C. Cooking processes at such 
heights are frequently accompanied by many difficul- 
ties. In a steam boiler, however, where the pressure is 
125 pounds per square inch, the boiling point of water 
reaches 170° C. 



HEAT: TRANSFERENCE AND TRANSFORMATION 243 

Solid substances dissolved in a liquid raise its boiling 
point. A saturated solution of common salt boils at about 
109° C. But tlie vapor rising from boiling brine is pure 
water vapor and condenses at 100° C. under an atmospheric 
pressure of 760 millimeters. 



Ether . . 
Chloroform 
Alcohol . 



Table of Boiling Points 
Pressure 760 millimeters 

. . 35° C. Benzine ...... 80° C. 

. . 61° C. Turpentine 160° C. 

. . 78.2° C. Mercury 350° C. 




248. Distillation. — When a liquid is vaporized in one vessel and 
the vapor afterwards condensed in another, the process is known as 
distillation. By this pro- 
cess pure water can be ob- 
tained from water contain- 
ing dissolved substances 
and other foreign matter. 
The liquid to be distilled is 
placed in a vessel A, Fig. 
179, and boiled. The va- 
por is conducted through 
the tube B, which is sur- 
rounded by a larger tube 
C containing a stream of 
cold water. The vapor is 
condensed on the cold walls of the small tube, and the resulting liquid 
runs out at the lower end into the vessel D. 

If two liquids are mixed together and heated in vessel A, the 
one having the lower boiling point will be vaporized first. Its vapor 
can be condensed and collected in a separate vessel D. Alcohol is 
thus separated from fermented liquors, and gasoline and kerosene 
from crude petroleum. 

249. Heat of Vaporization. — Fill a glass flask about half full 

of water, and place it over a flame to boil. Suspend one thermometer 
ill the liquid and another a little above it. While the water is boiling, 
read both thermometers. They will continue to read practically alike. 



Fig. 179. — Elustrating the Process of 
Distillation. 



244 



A HIGH SCHOOL COURSE IN PHYSICS 



Continue to apply heat until it is evident that the temperature of the 
water and steam is not raised above the boiling point. 

The experiment shows clearly that the heat applied to 
the flask is not utilized in raising the temperature of the 
water or of the steam. As in the process of melting a solid 
(§ 238), heat is here transformed from kinetic into poten- 
tial energy while changing the molecular condition of the 
water. For every gram of water vaporized a definite 
quantity of heat disappears. The amount of heat required 
to change a gram of any liquid at its boiling point into vapor 
at the same temperature is called the heat of vaporization ^ of 
that liquid. It represents the work that has to be done in 
producing a separation of the molecules of the liquid 
against their mutual attractions. 

When condensation., the reverse of vaporization, takes 
place, an amount of energy equal to the heat of vaporiza- 
tion is given up by the condensing vapor. Thus the 
quantity of heat required to vaporize a certain mass of 

water at 100° C. is all delivered up 
by the steam when it returns to the 
liquid state. 

250. Heat of Vaporization of Water 
Measured. — The heat of vaporiza- 
tion of water can be readily meas- 
ured by allowing a known mass of 
steam to condense in, and deliver 
its heat to, a known mass of water. 

Allow steam from a flask of boiling 

water, Fig. 180, to pass through a tube into 

a beaker containing, say, 400 grams of cold 

water. A trap T should be introduced in 
Measuring the i , • a j? j j. • j. xu 

Heat of Vaporization of «^^^^ *« ^°«^^^® ^ ^^^ ^^ ^^^ «*^^"^ i"*^ *h® 
Water. cold water. During the experiment note 




Fig 



1 Sometimes called " latent " heat of vaporization. 



HEAT: TRANSFERENCE AND TRANSFORMATION 245 

the initial and final temperatures of the water. From 5° C. to 35° C. 
is a good range over which to work. The mass of steam condensed 
is found by ascertaining the gain in the mass of water in the beaker 
during the process. 

For example, let the initial temperature of the water in an experi- 
ment be 5.6° C, the final temperature 35° C, and let the mass of 
water at the beginning be 400 grams, and at the close 419.5 grams. 

Let X be the heat of vaporization of water. 

The heat gained by the cold water = 400 (35 — 5.6) calories. 

The heat lost by the 19.5 grams of 

water which was formed by the 

condensed steam on cooling from 

100° C. to 35° C. = 19.5 (100 - 35) calories. 

The heat delivered by the steam at 

100° C. in changing to water at 

100° C. ' = 19.5 X calories. 

Hence, 19.5 x + 19.5 (100 - 35) = 400 (35 - 5.6) ; 

whence, x — 538 calories. 

The heat of vaporization of water accepted by physicists 
is 536 calories. 

251. Artificial Ice. — Certain substances that are gases under 
ordinary conditions of temperature and pressure become liquids tvJien the 
pressure is sufficiently increased. This is the case of ammonia gas, the 
gas that is given off from common aqua ammonia. The pressure re- 
quired to liquefy this gas under ordinary temperatures is about 10 
atmospheres, or 150 pounds per square inch. On the other hand, 
liquefied ammonia returns to the gaseous state when the pressure is 
reduced, and for each gram that vaporizes a quantity of heat equal to its 
heat of vaporization (§ 249) is abstracted from its surroundings. Upon 
these principles is based the operations of the artificial-ice machines 
in common use. 

An artificial-ice machine consists of three essential parts : (1) the 
compressor, (2) the condenser, and (3) the evaporator. See Fig. 181. 
The compressor is a pump which is run by an engine or motor whose 
function is to force ammonia gas under a pressure of about 10 atmos- 
pheres into the coils of the condenser C. Here the gas liquefies and 
gives up heat to the surrounding water which carries it away. From 
the condenser coils the liquefied ammonia passes through the regulat- 
ing valve V into the coils of the evaporator E, where the pressure is 



246 



A HIGH SCHOOL COURSE IN PHYSICS 



kept below two atmospheres by the continual removal of ammonia gas 
by the compressor. The rapid vaporization of the liquid ammonia 
under the reduced pressure in these coils causes it to take heat from 
the surrounding brine. By this abstraction of heat the temperature 
of the brine is reduced to a point several degrees below the freezing 
point of water. 




Fig. 181. — Artificial Cooling and Ice-making Apparatus. 

In the production of ice the evaporator E is so constructed that 
metal vats of pure water of the desired size can be lowered into the 
cold brine and left until frozen. These vats are then withdrawn from 
the brine, and the ice removed to some place of storage. 

The artificial cooling of storage rooms is brought about by cooling 
brine, as in the manufacture of ice. From the evaporator the cold 
brine is forced through coils of pipe placed about the walls of the 
rooms to be cooled. Inasmuch as there is no chance for the ammonia 
to escape, it can be used repeatedly with very little loss. The pres- 
sures are controlled by the regulating valve and may be read at any 
time from the gauges placed as shown in the figure. 



EXERCISES 

1. Explain the formation of moisture on the interior surface of 
windows. 

2. The temperature of blades of grass and leaves of trees falls 
rapidly on cloudless evenings. What has this to do with the forma- 
tion of dew? 

3. Does heating the air in a room remove the water vapor ? Why 
is the air in an artificially heated room usually " dry " ? 

Suggestion. — It is shown in § 245 that the dryness of the air does 



HEAT: TRANSFERENCE AND TRANSFORMATION 247 

not depend wholly upon the water vapor present in a given space. 
The student should try to write out in full the entire explanation. 

4. Heat a beaker of water over a flame, and observe that small 
bubbles rise to the surface long before the boiling point is reached. 
Compare this with the phenomenon of boiling. 

5. When steam is allowed to flow through a tube into cold water, 
a loud sound is produced. Explain. 

6. What becomes of the cloud that one sees near the spout of a 
teakettle ? Is it steam ? 

7. Will clothes dry more quickly on a still or a windy day? 
AVhy? 

8. How much heat is required to raise the temperature of 30 g. 
of water from 0° C. to 100° C. and convert it into steam? 

9. If 50 g. of steam at 100° C. change into water at 40° C, how 
much heat is given out ? 

10. If the heat delivered by 10 g. of steam in condensing at 
100° C. and cooling down to 0° C. were all applied to ice at 0° C, how 
many grams of ice would be melted? Ans. 79.5 g. 

11. A vessel containing 600 g. of water at 20° C. is heated until 
it is one half vaporized. How many calories have been received? 

12. The boiling point of water falls 1 centigrade degree for a de- 
crease in pressure of 2.7 cm. Find the boiling point when the 
barometer reads 74.5 cm. 

13. The boiling point of water falls 1 centigrade degree for an 
elevation of 295 m. above sea level. Find the temperature of boiling 
water at Denver, Colo., altitude 1600 m. above sea level. 

14. If water boils at 85.5° C. at the top of Mont Blanc, what is the 
altitude ? 

15. If 20 g. of steam at 100° C. are passed into 500 g. of water at 
5° C, what will be the resulting temperature? Ans. 29.2° C. 

16. How much heat will be required to convert 150 g. of ice at 
0° C. into steam at 100° C. ? 

2. THE TRANSFERENCE OF HEAT 

252. Three Modes of Heat Transmission. — Heat is trans- 
ferred from one point to another in three different ways ; 
viz. by conduction, convection, and radiation. By conduc- 
tion., heat passes through the metal walls of a stove or along 
a, metal rod from heated regions toward cold ones. By 



248 A HIGH SCHOOL COURSE IN PHYSICS 

convection^ the currents of air set up by a hot stove trans- 
fer heat to the distant parts of the room. By radiation^ 
energy from the sun reaches the earth ; and, by this pro- 
cess, the hands held before the fire in an open grate or 
near a hot stove become warm. 

253. Conduction. — The conduction of heat is simply the 
transference of molecular motion from those portions of a 
body Avhere such motion is greatest to those where the mo- 
tion is less, i,e, from the warmest parts of a body to colder 
parts, without producing any sensible motion in the inter- 
vening parts. The facility with which the conduction of 
heat takes place varies widely with the nature of sub- 
stances. 

Join together three similar bars of copper, brass, and iron as shown 
in Fig. 182, and heat the junction in a flame for several minutes. By 

sliding the tip of a sulphur match from the 
cold end of each bar toward the flame, ascer- 
tain at what point it ignites. In this manner 
it may be shown that the copper has conducted 
heat the farthest from the source and is, conse- 
quently, the best conductor. The iron will 
prove to be the poorest conductor of the three 
metals. 

Make tests by using a piece of glass tubing. 
Fig. 182. — Testing the the stem of a clay pipe, a piece of crayon, etc. 

Conductivity of These substances will be found to conduct very 

Metals. 1 

poorly. 

254. Liquids and Gases Poor Conductors of Heat. — The 

conductivity of a liquid can be tested as follows : 

Through a cork fitted in the neck of a glass funnel pass the tube 
of a simple air thermometer (Fig. 170), as shown in Fig. 183. Fill the 
funnel with water until the bulb is covered by about half an inch of 
the liquid. Pour about a spoonful of ether upon the water and ignite 
it. Although the flame is at all times separated from the bulb of the 
thermometer by only a thin layer of water, the liquid in the tube will 
remain stationary. 




HEAT: TRANSFERENCE AND TRANSFORMATION 249 



The experiment shows clearly that water is a very poor 
conductor of heat. Its conductivity is less than j-^-q that 
of copper. In general, all liquids, 
except mercury and other metals 
in a molten state, are to be classed 
as poor conductors. 

Gases have a lower conductivity 
than liquids. For this reason many 
substances, as wool, which inclose 
a large amount of air are poor con- 
ductors and are therefore used ex- 
tensively in the manufacture of 
winter garments. Such articles of 
clothing owe their warmth to the 
fact that they prevent the loss of 

the natural heat Fig. 183.— Testing the Con- 
Of the body. ductivity of Water. 

Winter wheat and fruit trees are pro- 
tected in a similar manner by a deep 
covering of snow. A flannel " holder " 
prevents the transference of heat from 
the flatiron to the hand, and a piece of 
ice wrapped in a woolen blanket is 
shielded from the heat of the atmos- 
phere. 

255. Convection. — Heat is trans- 
ferred in liquids and gases by the pro- 
cess of convection, i.e. by a general mass 
movement of the heated portions away 
from the source of heat. 





Fig. 184. — The Con- 
vection of Heat by 
Water. 



1. Pass the ends of a glass tube, bent as 
shown in Fig. 184, through a rubber stopper 
fitted to the neck of a bottle from which the 
bottom has been removed. Fill the apparatus 



250 



A HIGH SCHOOL COURSE IN PHYSICS 




with water, and place a small quantity of oak sawdust in the liquid to 
serve as an index of the motion. If a flame is moved back and forth 

between the points A and B, it 
will be seen that a current is set 
up in the direction from A to- 
ward B. In a short time the en- 
tire mass of water in the reservoir 
C will become hot. 

2. Make two openings in the 
side of a crayon box, as shown 
in Fig. 185. Set a short candle 
over one hole, leaving an opening 

^ ^„^ ^ . ^ . ,. into the box. Set a lamp chim- 

FiG. 185. — Convection Currents in Air. , , , -, ,, i ., 

ney over each hole and attach it 

to the box by means of melted candle wax. Light the candle and then 

hold some burning paper above chimney A . It will be observed that the 

flame and smoke of the burning paper will be drawn downward, thus 

showing the direction of the draft of air. At the same time heat is 

transferred upward from chimney B by the convection currents of the 

hot gases. 

256. Convection Explained. — Convection is brought 
about by the expansion of fluids (^i.e. liquids and gases) 
when heated. When a portion of the fluid is warmed, its 
volume increases, thus decreasing the density of the fluid. 
This portion of the body of the fluid is then forced upward 
in a manner similar to that in which a submerged piece of 
cork is forced up by water. (See § 121.) As the heated 
portions of the fluid rise, they carry their heat with them, 
and colder portions flow in to replace them. Convection 
currents are applied to the heating of buildings with hot 
air and hot water and to mine ventilation. The trade- 
winds and the Gulf Stream are convection currents of 
enormous proportions. 

257. A Hot-air Heating System. — Figure 186 shows the 
method employed in heating a house by means of hot air. 
A furnace is placed in the cellar and supplied with fresh 
air through the duct A leading to the heating chamber c?, e. 



HEAT: TRANSFERENCE AND TRANSFORMATION 251 



Here the air is heated and thence conducted through large 
pipes to the various rooms, of the building. A large part 
of the air thus led 
into the rooms finds 
an outlet around the 
windows and doors. 
Sometimes provision 
is made for its escape 
into a cold-air flue 
leading through the 
roof. A cold-air duct 
B is often introduced 
for the purpose of re- 
conducting the par- 
tially cooled air to 
the furnace, where it 
is again heated and 
sent into the rooms. 
For good ventilation, 
however, an abundant supply of fresh out-door air should 
be admitted to the system at A. The circulation of air 
is indicated by the arrows. The fire in the fire-box is 
controlled by dampers. These are often regulated by 
chains extending to a room above, and shown here by the 
dotted lines. 

258. A Hot- water Heating System. — A hot-water system 
of heating depends upon convection currents produced as 
shown in Experiment 1, § 255. Water is raised nearly to 
the boiling point in a heater H^ Fig. 187, placed in the 
basement. From the heater it is conducted through pipes 
to iron radiators R placed in the various rooms of the 
building, while the cooler water from the radiators is led 
in return pipes back to the heater. Thus a continuous 
current of water is maintained until the pipes are closed by 




Fig. 186. — Hot-air Heating System. 



252 



A HIGH SCHOOL COURSE IN PHYSICS 



valves placed near the radiators. On account of the large 
exposed surface in each radiator, the heat emitted by the 

hot water is transferred to 
the surrounding air. In or- 
der to prevent the radiation 
of heat from the conducting 
pipes, a thick covering of as- 
bestos, a very poor conductor 
of heat, is frequently pro- 
vided. This plan of heating 
affords no means of ventila- 
tion. 

259. Radiation. — When 
t WMVhVM P we stand before a hot stove 

L l^^^^^li'.it^ or a grate lull oi glowing 

coals, we readily perceive that 
the surface of the body near- 
est the fire is rapidly heated. 
Fig. 187. — Heating a House by However, the intervening air 

Means of Hot Water. . , , . , , , 

IS not warmed as it would 
be if the heat passed to us by conduction or convection. 
Again, a room is frequently heated by the sun's rays, while 
the glass through which the rays pass remains cold. 
Plainly, the medium which transmits the heat energy in 
these instances is not the air nor the glass. In order that 
the phenomena just described and others of a similar na- 
ture may admit of explanation, it is assumed by physicists 
that all space is filled with an exceedingly light medium 
called the ether. ^ The properties of the ether are such that 
transverse wave motions are transmitted by it in a manner 
somewhat similar to that in which the waves produced by 
a falling pebble are carried along upon the surface of 

1 The term must not be confused with •' ether, ^' which is a well-known 
liquid and is used in several experiments. 




HEAT: TRANSFERENCE AND TRANSFORMATION 253 

water. Energy thus transmitted is called radiant energy, 
and the process is called radiation. If this energy affects 
the sense of sight, it is called light. When it falls upon 
the hands, it produces warmth. Radiant energy becomes 
real heat only as it falls upon matter which is capable of ab- 
sorbing it and converting it into the energy of molecular 
motion. 

260. Absorption of Radiant Energy. — The ability of a 
body to radiate energy depends both upon its temperature 
and the nature of its surface. Smooth and highly polished 
bodies radiate poorly, while rough, black bodies radiate 
well. On the other hand, bodies differ in their power to 
absorb radiant energy. Those that radiate well also ab- 
sorb well. 

Nail two pieces of tin A and B, Fig. 188, to a block of wood as 
shown. Coat the interior surface of B with lampblack and attach a 
match to the exterior surface of each with 
melted paraffin. Now place a hot iron ball 
midway between the plates. In a moment 
the wax on B will soften, and the match will 
fall. 




• 




The experiment clearly shows that 
the blackened surface B absorbs the 
eners^y radiated by the ball faster than 

^, 1 . , ^ ^ TP • J? 1 1 1 Fig. 188. — The Black 

the bright one A. If pieces of black su^^ce B Absorbs 
and white cloth are placed upon snow, ^^^^ Better than the 

. c ^ T Polished Surface A. 

the rapid absorption oi the radiant 
energy of sunlight will cause the black body to melt 
its way into the snow. Since the white cloth reflects 
and transmits the greater portion of the energy, little re- 
mains to be converted into heat. This fact accounts for 
the general use of light-colored clothing in summer and, 
in part, for the stifling heat developed in attics under dark- 
colored roofing. 



254 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 189. — Crooke's 
Radiometer. 



261. The Radiometer. — This interesting instrument, 
Fig. 189, was invented by Sir William Crookes,i of Eng- 
land, in 1873. It is used to detect radiant 
energy. The instrument consists of a 
glass bulb from which the air has been 
almost exhausted and within which four 
diamond-shaped mica vanes are delicately 
pivoted on light cross arms. One face 
of each vane is coated with lampblack. 
When radiant energy falls upon these 
vanes, a rotation is produced. 

Since the blackened faces of the vanes 
absorb radiant energy, they are raised to 
a higher temperature than the bright 
faces. Thus the few remaining molecules of gas in the 
bulb have their speed greatly quickened as they 
come in contact with the black surfaces, and hence rebound 
from these faces with a strong reaction. It is this reaction 
that causes the vanes to move. The speed of rotation 
depends upon the intensity of the radiation falling upon 
the instrument. 

262. Selective Absorption of Bodies.— Place a radiometer 

near a lighted lamp and between them set a glass beaker. After the 

rate of rotation of the vanes has become uniform, fill the beaker with 

water. The rate of rotation will be greatly diminished. Repeat the 

experiment, but fill the beaker with carbon disulphide. It will be 

found to have httle effect on the motion of the vanes. Substitute a 

solution of iodine in carbon disulphide. Although nearly opaque to 

' light, the solution will be found to transmit the radiations perfectly. 

Water, which is transparent to the short, or visible, 

ether waves emitted by the lamp, transmits very poorly 

the longer waves of a slower rate of vibration. Likewise, 

glass transmits well the visible radiation (i.e. light) from 

the sun, but retards effectively the longer waves emitted 

1 See portrait facing page ^QQ. 



HEAT: TRANSFERENCE AND TRANSFORMATION 255 

by the objects in a room. Substances like glass and water 
which absorb long waves are called athermanous substances. 
While the glass in a window admits light energy into a 
room, the energy of the longer waves sent out from the 
heated objects within is retained. The glass of a green- 
house or hot-bed transmits well the energy of short waves 
to the soil within, but the longer waves emitted by the 
heated soil cannot escape. Hence the temperature rapidly 
rises. 

On the other hand, the carbon disulphide and the iodine 
solution transmit well the waves of the ether that are far 
too long to affect the sense of sight. Such substances are 
called diathermanous substances. 

263. The Sun as a Source of Heat. — The process of 
radiation plays an important part in everyday life, i? The 
sun is continually sending out great quantities of radiant 
energy in all directions in space. A small fraction of 
this energy falls upon the earth's atmosphere, passes read- 
ily through it without producing any appreciable change, 
and reaches the earth's surface. Here a large part of the 
energy of the ether waves is transformed into heat, i.e. is 
absorbed. The earth also radiates heat ; but being of a 
low temperature, the waves emitted by it are longer. 
Since the presence of water vapor in the atmosphere 
renders it athermanous, the radiation of energy away 
from the earth is greatly hindered. 

It is the radiant energy from the sun converted into 
heat that evaporates water, resulting in the production 
of vapor and rain. Rains produce the flow of rivers and 
thus give rise to the energy derived from waterfalls. The 
wood we use and the food we consume owe their value to 
the energy which they have stored up within them. This 
they derive from the sunlight and warmth in which they 
grow. Coal received its energy from the plants that 



V 



256 A HIGH SCHOOL COURSE IN PHYSICS 

flourished under the solar radiation of past ages. It is 
this energy that we utilize in warming our houses, cook- 
ing our food, and that we convert into mechanical energy 
through the help of the steam engine for running factories 
and aiding transportation over both land and water. 



EXERCISES 

1. In the construction of brick and cement houses the walls are 
often made hollow. Why ? 

2. Why does a piece of iron feel colder than a piece of wood when 
both have the same temperature ? 

3. What is the normal temperature of the blood on the centigrade 
scale ? of a living room ? 

4. Does woolen clothing supply the heat that maintains the tem- 
perature of the body ? 

5. Explain under what conditions a workman might be led to wear 
woolen garments to keep himself cool. 

6. Why are coverings of sheets of paper often sufficient to prevent 
plants from freezing on frosty nights? 

7. Why does dew seldom form on cloudy nights? Why are frosts 
almost entirely j^revented by the presence of clouds? 

8. In a tireless cooker a kettle containing vegetables that have been 
boiled a short time is surrounded by w^ool, felt, etc., and left a few 
hours to complete the process of cooking. Explain. 

9. The poor conductivity of glass causes a tumbler to crack when 
hot water is poured into it. Explain. Why does not a thin glass 
beaker crack from the same cause ? 



3. RELATION BETWEEN HEAT AND WORK 

264. Heat and Mechanical Energy. — The experiments 
made in section 212 show clearly that an intimate relation 
exists between heat and work. The production of heat at 
the expense of mechanical energy is one of the most com- 
mon phenomena of nature. In fact, the energy expended 
in nearly all mechanical processes passes finally to the form 
of heat. An inquiry into the reverse transformation of 




COUNT RUMFORD (SIR BENJAMIN THOMPSON) 

(1753-1814) 



The name of Rumford is prominent among the early physicists 
who engaged in confuting the theory that heat was a substance. 
His attainments as a soldier and public benefactor, howeyer, are of 
no less interest. 

Thompson was born on a farm near Woburn, Massachusetts, and 
as a young man was employed in teaching school. Haying been 
made a major in the local militia by the governor of New Hamp- 
shire, he became the object of mistrust by the friends of American 
liberty. On this account, in 1776 he removed to London. Here his 
advance was rapid; within four years he became undersecretary of 
state. In 1779 he was elected a member of the Royal Society. In 
1783 he planned to aid the Austrians against the Turks, but while 
on his way he met Prince Maximilian (afterwards elector of Bava- 
ria), who induced him to enter the Bavarian military service. For 
eleven years he remained at Munich as minister of war, minister 
of police, and grand chamberlain to the elector. He reorganized 
the Bavarian army, suppressed begging, provided employment for 
the poor, and established schools for the industrial classes. 

In 1791 Thompson was made a count of the Holy Roman Empire 
and took the name of Rumford. In 1799 he was instrumental in 
founding the Royal Institution of London and selected Sir Humphry 
Davy as the first lecturer. In remembrance of the help that he 
received in his early days from attending some lectures by Professor 
Winthrop at Harvard, Thompson later gave an endowment which 
founded the professorship that bears his name. His last years were 
passed near Paris, where he died in 1814. His tomb is at Auteuil. 



HEAT: TRANSFERENCE AND TRANSFORMATION 257 




Fig. 190. — Water raised 
by Heat Energy. 



energy, i.e. from heat into mechanical energy, is an inter- 
esting consideration. 

Let a tube, bent as shown in Fig. 190, project through a I'ubber 
stopper into a flask A half filled with water. The tube should extend 
nearly to the bottom of the vessel. Heat the 
water over a burner, and the water will be 
elevated into a tumbler at B. 

In this experiment work is per- 
formed upon the water, and heat is 
converted into potential energy., which 
is stored in the elevated water. 

265. Early Historical Experiments. 
— In the early development of the 
subject scientists looked upon heat as 
a kind of material substance called 
caloric. An important step toward ascertaining the true 
nature of heat is found in the experiments of Count Rum- 
ford. ^ He showed that one horse used as a source of power 
could develop sufficient heat by friction to raise 26.5 pounds 
of water from the freezing to the boiling point in 21 hours. 
About this time Sir Humphry Davy (1778-1829) showed 
that two pieces of ice kept below the freezing point could be 
melted by rubbing them together. The first, however, to 
establish the relation between heat and work by expressing 
one in terms of the other was James Prescott Joule ^ 
of Manchester, England. 

266. Joule's Experiment. — The method employed by 
Joule to ascertain the exact relation between the calorie 
and the unit of mechanical energy consisted in measuring 
the heat produced by a definite quantity of work. The 
heat under consideration was produced by the rotation of 
paddles in a vessel of water 6', Fig. 191. The work 
done upon the water produced heat enough to raise its 



1 See portrait facing page 256. 
18 



2 See portrait facing page 258. 



258 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 191. — Illustrating Joule's Method for 
Determining the Mechanical Equivalent 
of Heat. 



temperature an appreciable amount. The rise in tempera- 
ture being measured, the quantity of heat developed could 

be computed as the 
/^ product of the mass of 
water and its change 
in temperature (§ 231). 
The paddles were 
turned by two weights 
TF, W, attached to cords 
so arranged as to ro- 
tate the main shaft. 
The work performed 
by the paddles could be computed as the product of the 
weights and the distance through which they descended 
(§ b5^. Thus the work corresponding to a calorie of heat 
could readily be determined. 

267. The Mechanical Equivalent of Heat. — Joule's ex- 
periments upon the relation of heat and work extended 
over more than one half his life. They have been care- 
fully repeated (1879) by Professor Rowland of Johns 
Hopkins University, with apparatus of more refinement 
and precision. As a result of these experiments the follow- 
ing value of the calorie is generally accepted by physicists: 

1 calorie = 427 gram-meters^ or 41,900,000 ergs. 

This result is known as the mechanical equivalent of 
heat, or simply Joule* s equivalent. 

268. Conversion of Energy. — The experiments per- 
formed by Rumford, Joule, Rowland, and others, relative 
to the conversion of mechanical energy into heat, serve to 
verify the principle of the Conservation of Energy first 
stated in § 64. Ignorance of this great law of nature has 
led men at all times, and even in this enlightened period, 
to undertake to construct devices whereby useful work 




JAMES PRESCOTT JOULE (1818-1889) 



The attention of Joule was turned at an early age in the direc- 
tion of physics and chemistry by the influence of his teacher, John 
Dalton, the chemist. Under his tuition, Joule was initiated into 
mathematics and trained in the art of experimentation. His renown 
rests upon the thoroughness with which he established the doctrine 
of the Conservation of Energy (§ 64) upon an experimental basis. 
His epoch-making experiments were made to determine the mechan- 
ical equivalent of heat, which is recognized as one of the most im- 
portant physical constants. Measurable quantities of energy were 
expended in revolving paddles in water, mercury, and oil, and cast 
iron disks were rotated against each other and the resulting quantity 
of heat ascertained. The results of these experiments left no dovibt 
that the amount of heat prodviced by a definite quantity of mechan- 
ical work is fixed and invariable. For this great scientific achieve- 
ment Joule received the Royal Medal of the Royal Society of 
England in 1853, and eight years later, when men of science more 
fully understood the value of the discovery, was presented with the 
Copley Medal. 

Joule was the son of a wealthy brewer of Manchester, England, 
at which place he carried on his experiments. Important laws 
relating to the heating of electrical conductors and valuable con- 
tributions to the subject of electro-magnetism must also be accred- 
ited to him. The joule, which is used as a unit of energy, has been 
so named in his honor. 



HEAT: TRANSFERENCE AND TRANSFORMATION 259 

can be obtained without the expenditure of an equivalent 
amount of energy of some form. Such devices, if possible, 
would supply enough energy to keep their parts in motion 
when once started, and are therefore called perpetual-motion 
machines. Any attempt, however, to secure useful energy 
from the wind, sunlight, ocean waves, tides, etc., is praise- 
worthy, and should be encouraged. To some extent, such 
efforts have not been fruitless. But since in the best ma- 
chines that can be made some friction will exist, there will 
be a continual conversion of a part of the energy supplied 
to the machine into heat. If, therefore, energy is supplied 
only in starting the machine, no matter how it is con- 
structed, it is sure to come to rest. Consequently, a ma- 
chine at best can only transfer or transform the energy 
with which it is supplied. 

The transformation of mechanical energy into heat is 
easily accomplished. For example, a bullet is warmed by 
its impact with a target, mercury can be warmed by being 
shaken vigorously in a bottle, a mass of shot will rise in 
temperature if allowed to fall several feet, a button grows 
hot when rubbed upon a piece of flannel, etc. The reverse 
transformation, i.e. from heat into mechanical energy, is 
not, however, so readily effected. The process, neverthe- 
less, is accomplished in steam and gas engines by making 
use of certain properties of gases. 

269. Gases Heated by Compression and Cooled by Expan- 
sion. — We have seen in § 212 that the energy used in com- 
pressing a gas is converted into heat. On the other hand, 
when a gas is alloiued to expand agaifist pressure and perform 
work., heat is given up and the temperature of the gas falls. 
In other words, molecular energy is expended by the gas 
when it does work. In order to utilize this process in 
converting heat into useful work, the gas is allowed to ex- 
pand in a cylinder and thus move a piston P, Fig. 192. 



260 



A HIGH SCHOOL COURSE IN PHYSICS 



It is clear that, in the case illustrated, the gas performs 
work in raising the weight TF to a higher position. A 
modification of this process is employed in all 
steam and gas engines. 

270. The Reciprocating Steam Engine. — The 
essential parts of an ordinary steam engine are 
the cylinder, the piston, and the slide-valve 
mechanism, represented diagram matically in 
Fig. 193. The gas employed is steam gener- 
ated by the combustion of fuel. A to-and-fro 
motion is given to the piston P by the force 
Fig 192.— By exerted by the steam which is applied to its 

Movmga 

Piston p a two faccs alternately. The operation is as fol- 

Gas Does^Q^g. 

Work on 

the Weight Steam under a pressure of several atmospheres (i.e. 
100 to 250 pounds per square inch) enters the steam 
chest S from the boiler. From S the steam finds an entrance into the 
cylinder through the port N and drives the piston to the left, forcing 
any gas that may be contained in the space C 
through the port M and out of the engine through ^^^^^SM^S^z^z 





Section of the Steam 
Engine. 



the exhaust pipe E. The motion of the piston is communicated to the 
main shaft A through the connecting rod R and the crank D. As the 
piston approaches the left end of the cylinder, the sliding valve V is 
moved to the right by the eccentric F and the eccentric rod E', thus 



HEAT: TRANSFERENCE AND TRANSFORMATION 261 

admitting " live " steam through M into the cylinder chamber C ', and 
opening the port N to allow the expanded steam to escape. The 
piston is now driven back to the right, and the sliding valve V is forced 
back to its former position just before the piston reaches the end of 
its stroke as at first. The operations just described are then repeated. 
The shaft of the engine is provided with a heavy fly wheel E in order 
to maintain uniformity of speed. 

In the so-called non-condensing or high-pressure engines the exhaust 
steam escapes through the exhaust pipe E into the open air. The pis- 
ton of such an engine therefore moves continually in opposition to the 
pressure of the atmosphere. This disadvantage is partially removed 
in condensing engines. In engines of this type the exhaust pipe E 
conducts the exhaust steam to a condensing chamber in which a 
spray of cold water hastens its condensation. By the aid of a pump 
operated by the engine, the water together witli the condensed steam, 
is removed from the condensing chamber, leaving a back-pressure of 
only a few ounces per square inch instead of one atmosphere. Thus 
by decreasing the back-pressure against the piston, a larger quantity of 
useful work can be obtained from the steam, and the efficiency of the 
engine correspondingly increased. Condensers are not used on loco- 
motives (1) because of the large supply of cold w^ater necessary and 
(2) because of their inconvenient size. 

271. The Gas Engine. — With the development of the automo- 
bile has come that of the gas engine as a source of power. To this 
class of machines belong all engines utilizing an explosive mixture of 
gases as the working agent, such as air and illuminating gas, or air 
and gasoline vapor. The operation of the so-called " four-cycle " gas 
engine in common use is shown in Fig. 194. 

C is a cylinder within which moves the piston P. The piston is 
connected with the crank B by means of the rod A. Upon the crank 
shaft D is mounted a heavy fly wheel IF, wliicli is set in motion by 
the hand on starting the engine. When the piston moves downward 
to the position shown in (1), an explosive mixture of gas and air is 
drawn into the cylinder through the inlet valve /. As the motion 
continues, the piston moves upward and compresses the mixture in 
the top of the cylinder, as shown in (2). At about the instant the 
piston reaches the highest point, as in (5), an electric spark at the 
spark-plug 6" ignites the gas ; an explosion ensues, with the produc- 
tion of much heat, and the expanding gases exert an enormous pres- 
sure on the top of the piston. This forces the piston violently down- 



262 



A HIGH SCHOOL COURSE IN PHYSICS 



ward, giving motion, and hence kinetic energy, to the heavy fly wheel 
W. On the next upward stroke the products of the combustion of the 
gas are driven out of the cylinder through the exhaust valve, as shown 
in (4). This valve is opened automatically at the proper instant. 
The piston having traversed the length of the cylinder four times, the 




0) (^) <^J r^J 

Fig. 194. — Operation of a Four-cycle Gas Engine. 



initial conditions are restored, and the operations are repeated. It is 
obvious that the fly wheel must be made heavy, since the energy given 
it during the third stroke of the piston has to keep the engine and ma- 
chinery in motion with almost constant speed during the three fol- 
lowing strokes of the cycle. 

When gasoline is used as fuel, the inlet pipe / leads from the " car- 
buretor," into which the liquid enters as a spray, vaporizes, and is 
mixed with the proper amount of air. In order to prevent undue 
heating of the cylinder and piston, a current of water is kept in circu- 
lation through cavities cast in the walls of the cylinder. Some manu- 
facturers supply so-called " air-cooled " engines, the cylinders of which 
are cooled by the circulation of air about their exterior surface. In 
this instance the cylinder is cast with numerous projections for the 
purpose of increasing as much as possible the amount of radiating 
surface. 



HEAT: TRANSFERENCE AND TRANSFORMATION 263 

On account of the lightness and compactness of the engine, and the 
small space occupied by the fuel, gasoline engines are extensively used 
to propel automobiles, in which motors of from one to six cylinders 
may be seen. Such engines are also widely used in launches, dirigible 
balloons, aeroplanes, pumping stations, machine shops, factories, etc., 
on account of the small attention required in their operation. 

272. The Steam Turbine. — In the common reciprocating form 
of the steam engine a large amount of energy is lost in stopping and 
starting the piston and connecting rods at the end of each stroke. It 






7J» c Xrr' 



B D 



F 



(2) 



Fig. 195. — Principle of the Steam Turbine. 



is only within the last few years that inventors have succeeded in 
designing efficient engines of the purely rotary type. The operation 
of a steam turbine is as follows : 

Steam under high pressure is conducted through aseries of stationary 
jets A, (i). Fig. 195, arranged in a circle, which directs it obliquely 
against a series of blades B which are attached to a rotating drum, 
called the 7~otor. The rotor is fastened to the main shaft of the engine. 
By the impact of the steam these movable blades are impelled in an 
upward direction and thus produce rotation. (^), Fig. 195, shows the 
arrangement of several series of movable and stationary blades used 
in the more powerful turbines. Each movable blade is a curved pro- 
jection attached to the exterior surface of the rotor ; each stationary 
blade is fastened to the interior surface of the metal case surrounding 
the rotor. The concave surfaces of the two sets of blades are turned 



264 A HIGH SCHOOL COURSE IN PHYSICS 

opposite to each other as shown. The number of series used will vary- 
largely in turbines of different power. After the steam has passed 
through the first series of movable blades B, a series of blades C, 
which are stationary, serves to direct it at the proper angle against 
the next series of movable blades D, and so on through the entire 
turbine. 

The steam turbines will find extensive use in ocean-going steamships 
on account of the fact that they are free from the objectionable vibra- 
tion that always accompanies engines of the reciprocating form. At 
the present time they are replacing the ordinary steam engine in the 
generation of electrical power, and in a few years, no doubt, will be 
found wherever energy is to be derived from steam. 

EXERCISES 

1. Explain how the energy contained in coal can be utilized in 
performing work. 

2. What energy other than that of coal is often employed in run- 
ning factories, etc. ? 

3. The heat developed by the combustion of a gram of coal of a 
certain grade is 5000 calories. How many kilogram-meters of work 
could be done if all the energy could be used for this purpose ? 

4. If all the potential energy stored in a 500-kilogram mass of 
rock at an elevation of 200 m. were converted into heat, how many 
calories would be produced ? 

5. The energy of a falling body is transformed into heat when it 
strikes. Compute the number of calories of heat produced when a 
10-gram mass of iron falls 25 m. 

Suggestion. — Compute in gram-meters the energy of the given 
mass at an elevation of 25 m., then reduce to calories. 

6. A steam engine raises 8000 four-pound bricks to the top of a 
building 75 ft. high. How many calories of heat are thus expended ? 
If the efficiency of the engine is 10 % (§ 100), how many calories must 
be developed by the combustion of the coal that is used? 

7. How high could 100 g. of ice be elevated by the amount of heat 
required to melt the same amount, if all the heat could be utilized for 
that purpose? 

8. Show that the energy required to vaporize 1 g. of water at 
100° C. is equivalent to the work done by a force of 10 kg. in moving 
a body a distance of 22.89 m. in the direction of the force. 

9. Show that it requires more energy to raise the temperature of 



HEAT: TRANSFERENCE AND TRANSFORMATION 265 

100 g. of iron from 0° C. to 100° C. than to elevate a weight of 400 kg. 
through a height of 1 m. 

10. The average pressure of the steam in the cylinder of an engine 
is 125 lb. to the square inch and the area of the piston is 50 sq. in. 
Compute the work done by the steam during each 20-inch stroke of the 
piston. 

11. If the mechanical equivalent of the calorie defined in § 231 is 3.1 
foot-pounds, compute the heat units lost by the steam in Exer. 10 at 
each stroke of the piston. 

12. At M'hat rate does a 2-horse-power engine consume coal when 
working at its full capacity, if its efficiency is 10 % and the coal pro- 
duces 5500 calories per gram ? 

SUMMARY 

1. The state of a body depends largely on its temper- 
ature. The temperature at which a solid changes to a 
liquid is its melting or fusing point. Definite laws gov- 
ern the fusion of all crystalline bodies (§§ 234 to 236). 

2. The process of liquefaction is accompanied by an 
absorption of heat. Molecular kinetic energy (heat) is 
converted into molecular potential energy in the process. 
The heat of fusion of a substance is the number of calories 
per gram required to liquefy it without changing its tem- 
perature. The heat of fusion of ice is 80 calories. Non- 
crystalline substances when heated pass through a plastic 
state and have no definite melting point (§§ 237 and 238). 

3. When a liquid solidifies, an amount of heat equal 
to the heat of fusion is given up for each gram (§ 239). 

4. Evaporation may take place at any temperature. 
It is a process by which many of the more rapidly moving 
molecules of a body become detached and pass into the 
surrounding space. The gas composed of these detached 
molecules is called a vapor (§§ 240 and 241) . 

5. The rate of evaporation of a liquid varies with the 
amount of exposed surface and its temperature, and is de- 



266 A HIGH SCHOOL COURSE IN PHYSICS 

creased by the presence of the vapor of the liquid in the 
space around it (§ 242). 

6. When the number of molecules leaving a liquid per 
second equals the number reentering it, the vapor is satu- 
rated. Its pressure at this time is called the maximum va- 
por pressure at that temperature (§ 243). 

7. The quantity of a given vapor required to produce 
saturation in a given space is the same whether the space 
is occupied by other vapors or not. This quantity de- 
pends, however, on the temperature (§§ 243 and 244). 

8. On account of the abundance of water, the atmos- 
phere always contains more or less water vapor. The 
temperature to which air would have to be reduced to 
cause moisture to form is called its dew-point (§ 245). 

9. The relative humidity of the air is the ratio of the 
amount of water vapor present in a given volume to the 
amount required to produce saturation at that tempera- 
ture (§ 245). 

10. Every liquid has its own boiling point, which is in- 
variable under the same conditions. This point rises or 
falls according as the pressure upon the liquid is increased 
or decreased (§§ 246 and 247). 

11. The heat of vaporization of a liquid is the number 
of calories required to convert 1 g. of it at its boiling point 
into vapor at the same temperature. The heat of vapori- 
zation of water is 536 calories. This amount of heat is 
given up by the vapor when it condenses (§§ 249 and 250). 

12. Heat is transferred by conduction^ convection, and 
radiation. Substances are classed as good and poor con- 
ductors of heat. In general, liquids (except liquid metals) 
and gases are poor conductors (§§ 252 to 254). 

13. Liquids and gases transfer heat by a general mass 



HEAT: TRANSFERENCE AND TRANSFORMATION 267 

movement of the heated portions away from the source of 
heat, i.e. by convection. Convection is the result of the 
expansion which accompanies a rise in temperature. 
Heating systems depend on the convection of heat by air, 
steam, or hot water (§§ 255 to 258). 

14. Radiation is the process in which energy is trans- 
ferred by ether waves. The energy of these waves is trans- 
formed into heat when absorbed by bodies. The best 
radiators and absorbers are rough, black bodies. The sun 
is our great source of energy. This energy we receive 
through the process of radiation (§§ 259 to 263). 

15. Heat may be made to perform work. One calorie 
is equivalent to 427 gram-meters, or 41,900,000 ergs. This 
result is known as Joule's equivalent^ or the mechanical 
equivalent of heat (§§ 264 to 267). 

16. Heat energy is transformed into mechanical energy 
in steam and gas engines. A gas does work in expanding 
against a movable piston. As a result of the work done, 
the temperature of the working gas is lowered (§§ 268 
to 272). 



CHAPTER XIII 
LIGHT: ITS CHARACTERISTICS AND MEASUREMENT 

1. NATURE AND PROPAGATION OF LIGHT 

273. Meaning of the Term "Light." — Just as sound is 
defined as undulations in the air, or some other medium, 
that produce the sensation which we call " sound," so lights 
in the same sense, consists of undulations or waves in the 
ether that produce the sensation which we often call by 
the name "light." (See § 259.) Not all ether waves can 
be regarded as light waves, since not all affect the organ 
of sight; but all ether waves, from the longest to the 
shortest, transfer energy, and therefore may properly be 
classed as carriers of radiant energy. 

274. The Ether and Ether Waves. — The theory that 
light is wave motion in the ether was advocated by the 
Dutch physicist Huyghens (1629-1695) in 1678. The 
theory, however, was not well established until the begin- 
ning of the nineteenth century, when the experiments by 
Thomas Young of England and Fresnel of France placed 
it on a firm basis. Ether fills all interstellar space as well 
as the spaces between the molecules in bodies of matter. 
The ether is also of extreme rareness, or tenuity, since 
planets passing through it suffer no appreciable retarda- 
tion in their orbits. 

Ether waves possess several well-known characteristics. 
They are transverse waves and are propagated with a defi- 
nite speed, and this speed becomes less when they pass 
through matter such as glass, air, water, etc. Ether 
waves may be reflected, transmitted, bent from their 

2G8 



LIGHT: CHARACTERISTICS AND MEASUREMENT 269 



courses, or their energy may be transformed into other 
forms than radiant energy. 

Ether waves that produce an effect upon the sense of 
vision vary in length between about 0.00004 and 0.00008 
centimeter. Hence our sense of sight, with its narrow lim- 
itations, does not enable us to perceive directly ether waves 
which are shorter than the former of these two numbers or 
longer than the latter. 

275. Speed of Light. — An achievement of great scien- 
tific importance was the discovery that light travels with a 
definite speed. Previous to the year 1676 it was supposed 
that light moved infinitely fast, because no one had found 
a way to measure so great a velocity. In that year the 
Danish astronomer Roemer (1644-1710), as the result of 
several months' work with the instruments at the Observa- 
tory of Paris, correctly inferred that the time required for 
light to traverse the diameter of the earth's orbit (about 
186,000,000 miles) Avas almost 1000 seconds. Roemer was 
led to this conclusion 
after making a series 
of observations upon 
the eclipses of one of 
the satellites of the 
planet Jupiter. At 
each revolution of the 
satellite s, Fig. 196, 
in its orbit around 
Jupiter J", it passes 
behind that planet 
into its shadow and becomes invisible from the earth at 
E. By measuring the interval between two successive 
eclipses of the satellite it was apparently possible to pre- 
dict the precise time of each eclipse for many months in 
advance. But when the earth was at E' ^ on the opposite 





Fig. 196. — Roemer's Method for Determining 
the Speed of Light. 



270 A HIGH SCHOOL COURSE IN PHYSICS 

side of the sun, the time of each eclipse of the satellite was 
found to be about 1000 seconds later than predicted. In 
order to account for this difference, Roemer advanced 
the idea that this interval was precisely the time that 
light requires to pass over the diameter of the earth's 
orbit. On this assumption the speed of light is 186,000,000 
-f- 1000, or 186,000 miles per seco7id. 

Recent measurements of the speed of light by different 
methods continue to show that it is about 186,000 miles, or 
300,000 kilometers, per second. 

EXERCISES 

1. The circumference of the earth is about 25,000 mi. How many 
times could this distance be traversed by light in a second? 

2. The distance of the north star from the earth is so great that it 
requires about 43 yr. for its light to reach the earth. Express the 
distance in miles. 

3. How many minutes are required for light to reach the earth 
from the sun ? 

4. Since the sun moves (apparently) through 360° in 24 hr., over 
what arc will it move while a light wave is on its way from the sun to 
the earth ? 

2. RECTILINEAR PROPAGATION OF LIGHT 

276. Light Travels in Straight Lines. — Set up a small screen 
about midway between a candle flame and the wall so that it casts a 
well-defined shadow. Mark the edge of the shadow, and extinguish 
the candle. Stretch a cord between the candle wick and the line mark- 
ing the edge of the shadow, and it will be found to graze the edge of 
the screen. Since the cord is straight, the course taken by the light 
from the candle to the wall is a straight line. 

Further evidence regarding the fact that light follows 
straight lines may be obtained by observing the path taken 
by a beam of light as it enters a partially darkened room 
where the air contains dust. We unconsciously utilize 
this important fact in many ways. In order that we may 
see an object, light must come from that object to the eye ; 



LIGHT: CHARACTERISTICS AND MEASUREMENT 271 

and we always assume that the object sending the light to us 
is located in the straight line which marks the direction of 
the light as it enters the eye. The marksman trains his gun 
along the line of direction of the light which comes from 
the object he wishes to hit, and the carpenter selects a 
straight piece of lumber by " sighting " along its edge. 

We shall see later, however, that light deviates from a 
straight line under certain conditions, but that the devia- 
tion is ordinarily inappreciable. 

277. Shadows. — A shadow is a space from which the light 
from a luminous body is wholly or partially excluded by an 
opaque body. The nature of a shadow depends both upon 
the form of the opaque body and upon the form of the 
source of light. 

1. Hold an opaque body, as a book, between a very small source 
of light, as an electric arc light, and a white w^all or screen. A very 
sharply outlined shadow will be produced upon the screen for all posi- 
tions of the opaque body. 

2. Place two electric arc lights about 15 centimeters apart in a 
line parallel to a screen or wall, and produce a shadow, as in Experi- 
ment 1. Two portions of the shadow are now easily distinguished, 
viz. a dark central part and a partially illuminated area just outside. 
The experiment may be performed with a single gas flame or by using 
two oil lamps placed a few centimeters apart. 

When the source of light L, Fig. 197, is small, and an 
opaque body AB intercepts the light, a region of darkness 




Fig. 197. — Illustrating the Shadow Cast by a Sphere. 

ABCD is produced behind it as shown by the shading. 
This space is the shadow of AB. It is obvious that the 



272 A HIGH SCHOOL COURSE IN PHYSICS 

form of the shadow may be found by drawing straight lines 
from L just touching the edge of the object AB, If AB 
is a sphere, it is clear that the form of the shadow will be 
that of a truncated cone whose top rests against the 
sphere. 

When two sources of light L and L\ Fig. 198, are 
used, no light from the source L enters the region CABD\ 




Fig. 198. — Showing the Production of Umbra and Penumbra. 

and none from L' enters the space C^ABB. Now the 
space CABB lies within both of these spaces and hence 
receives no light from either source. The remaining shaded 
portions receive light from one or the other of the sources. 
Furthermore, if other luminous points exist between L and 
X', the space CABB will receive no light from any of them. 
The portion of a shadow that is ivholly dark is called the 
umbra, and the portions that are only partially illuminated 
are called the penumbra. 

278. Eclipses Produced by Shadows. — In the preceding 
section it was seen that the section of a shadow that falls 
upon a wall or the ground will have a distinct outline 
only when the source of light is small. If, therefore, the 
source of light is the sun, we find no sharply defined 
shadows, i.e. every shadow is surrounded by an indistinct 
region which is partially illuminated. 

Let the sun. Fig. 199, be the source of light, and the 
earth the opaque body. By drawing lines tangent to 
both sun and earth, as shown, we find that the earth casts 



LIGHT: CHARACTERISTICS AND MEASUREMENT 273 




Fig. 199. — Showing the Moon Eclipsed by- 
Entering the Earth's Umbra. 



a shadow of which the umbra is cone-shaped and has its 
apex at A. Surrounding this is the penumbra which 
varies from total dark- 
ness near the umbra to 
practically full illumi- 
nation near its outer 
limits. If the moon 
Min its monthly revo- 
lution about the earth, 
in the orbit shown by the dotted line, passes completely into 
the earth's umbra, it receives no light from the san and is 
thus eclipsed. Since we see the moon only by the light 
which it reflects from the sun to the eye, this phenomenon 
constitutes a total eclipse of the moon. But if only a por- 
tion of the moon enters the earth's umbra, it suffers only 
a partial eclipse. 

279. Pin-hole Images. — Images are readily produced by means 
of small apertures. If a hole 2 or 3 millimeters in diameter is made 
in the window shade of a darkened room, images of trees, clouds, and 
other outdoor objects will be produced on a screen held a short dis- 
tance from the opening. Each dimension of an image is proportional 
to the distance from the aperture to the screen, but the larger the im- 
age, the less distinct it is in 
every detail, since the light 
is distributed over a larger 
area. 

The reason for the for- 
mation of images in this 
manner is made clear in 
Fig. 200. CD is a candle, 
A an opaque piece of wood 
or cardboard having a small 
aperture at iJ, and 5 is a 
white screen. Light from 
the tip of the candle C, 

Fig. 200. — An Image Produced by a Small fo^' example, falls at all 
Aperture. points on A, but only that 

19 




274 A HIGH SCHOOL COURSE IN PHYSICS 

falling at H is transmitted. This portion follows the straight line 
CH to F. Likewise, only light from D can fall at E on the screen. 
Thus the portions of light from the several points of the object CD 
build up the inverted image EF. 

Numerous images of the sun may often be observed upon the side- 
walk when the light passes through the small openings between the 
leaves of a tree. These images assume interesting, crescent-shaped 
figures during a partial eclipse of the sun. 

EXERCISES 

1. Hold a book in direct sunlight, and from the section of the 
shadow that is outlined upon the floor, infer whether we should treat 
the sun as a point source of light. Describe the shadow. 

2. Hold a ball in direct sunlight about 5 ft. from the floor or wall, 
and ascertain whether or not it casts a distinct shadow. Do the same 
beneath an uncovered electric arc light. Draw figures to illustrate the 
difference in the two shadows. 

3. Describe the shadow cast by the moon. In what direction does 
its umbra point ? Does its umbra ever reach the earth ? 

4. When the moon enters the earth's umbra, is its darkening 
gradual or sudden? Explain. 

5. If the earth should pass into the moon's umbra, what phenom- 
enon would be observed by a person standing in the shadow ? Would 
any of the sun be visible? Would any of the sun be visible to a per- 
son standing in the moon's penumbra ? 

Suggestion. — Draw a figure representing sun, moon, and earth 
in such a position that the umbra of the moon just touches the earth. 

6. If the sun's rays make an angle of 4.5° with the horizontal plane, 
how long is the shadow cast on level ground by a vertical pole 50 ft. 
high? 

7. A vertical rod 10 ft. in height casts a shadow 12 ft. long on a 
level sidewalk. How tall is a tree whose shadow at the same time is 
72 ft. in length ? 

8. How could one find the height of a building by employing the 
method suggested by Exer. 7 ? 

3. INTENSITY AND CANDLE POWER OF LIGHTS 

280. Intensity of Illumination. — If one realizes that the 
waves of light that are sent out from any given source 



LIGHT: CHARACTERISTICS AND MEASUREMENT 275 

spread out in all directions, it is readily inferred that the 
intensity of illumination will decrease as one recedes from 
the luminous body. We are also led to the same conclu- 
sion by the fact that we decrease the distance from a lamp 
to a printed page when we wish to increase the amount of 
illumination. The exact law is readily shown by experi- 
ment. 

Cut in a large cardboard screen A, Fig. 201, an aperture just 2 
inches square. Place the screen 1 meter from a point source of 




Fig. 201. — The Intensity of Light Varies Inversely as the Square 
of the Distance. 

light, preferably an electric arc. Now place a second screen B, upon 
which is drawn a square precisely four times as large as the aperture 
in A, i.e. 4 inches square, 2 meters from the light. The light, which 
at a distance of 1 meter falls upon an area A, at a distance of 2 
meters is found to cover precisely 4 equal areas. Hence each area at 
B receives only one fourth as much light as a similar area at A. 
When the second screen is carried to C, a distance of 3 meters from 
L, the light which passes through A illuminates 9 equal areas at C. 
Hence each area at C receives only one ninth as much light as an 
equal area at A. 

It is now plain that when the distance from a source of 
light is doubled, the intensity of illumination is divided 
by 4 ; and when the distance is made three times as great, 
the intensity of illumination is ^. Hence the experiment 
leads us to the conclusion that the inteyisity of illumina- 
tion is inversely proportional to the square^ of the distance 
from the source of light. 

281. Candle Power of Lights. — The law of intensity 
shown in the preceding section is used to compare the 



276 



A HIGH SCHOOL COURSE IN PHYSICS 




illuminating powers of two sources of light. If the 
intensity of one of the lights is known, that of the other 
can be found. 

Place a lighted candle A, Fig. 202, 1 meter from a paper screen *S 
and four similar candles at a point B, the same distance on the oppo- 
site side of S. It is now clear that the side of the screen facing the 
4 caudles receives 4 times as much illumination as the other. But 
the two illuminations may be equalized by moving the 4 candles to a 
greater distance. If, now, a drop of oil or candle wax is placed on the 
paper screen, it becomes possible to ascertain when the illuminations 
are equal, since the spot will look alike on the two sides when viewed 

at the same angle. To pro- 
duce equal illumination on 
the two sides of S (i.e. to di- 
vide the illumination pro- 
duced by the stronger light by 
4), it will be found necessary 
to move the 4 candles to a dis- 
tance of 2 meters from the 
screen. Hence the light-pro- 
ducing powers of the two lights are directly proportional to the squares 
of their respective distances from the screen. 

It is clear that this method may be employed in the 
comparison of the light-emitting powers of two sources. 
The process is to set the lights so that they illuminate 
the two sides of a screen equally; then the ratio of the 
squares of their respective distances from the screen ex- 
presses the ratio of the intensities of the two lights. A screen 
upon which is an oiled spot is used in the Bunsen photo- 
meter for the measurement of the power of lights. 

The unit used in the measurement of the power of 
lights is called a candle power and is approximately the 
power of a sperm candle of the size known as " sixes " 
(meaning six to the pound), burning 120 grains per hour. 

The candle power of a Welsbach gas lamp consuming 
about 3 cubic feet of gas per hour is from 50 to 100, and 



Fig. 202. — Showing a Method of MeaS' 
uring the Candle Power of Lights. 



LIGHT: CHARACTERISTICS AND MEASUREMENT 277 

that of ordinary open gas tiames is from 15 to 25, while the 
consumption of gas is from 5 cubic feet per hour upward. 
The incandescent electric lamps containing a carbon fila- 
ment in most common use are of 16 candle power, but 
those of greater power can be procured. 

EXERCISES 

1. A 2-caridle-power light is placed 1.5 m. from a screen. Where 
must an 8-candle-power light be placed to produce the same illumina- 
tion on the screen ? 

2. In measuring the candle power of an electric light it was found 
that a 4-candle-power light placed 2 m. from a disk produced the same 
illumination as the electric light at 10 m. Compute the power of the 
electric light. 

3. If a book receives ample illumination when placed 10 ft. from 
a 50-candle-power lamp, how far must it be placed from a liglit of 
5-candle-power to be equally well illuminated? 

SUMMARY 

1. Light, physically speaking, consists of ether waves 
which produce the sensation called lights i.e. which excite 
the optic nerve (§§ 273 and 274). 

2. The speed of light is about 186,000 miles (300,000 
km.) per second (§ 275). 

3. Light is propagated in straight lines in a uniform 
medium. This fact gives rise to shadows, eclipses, pin- 
hole images, etc. (§§ 276 to 279). 

4. When a luminous body is of appreciable size, the shad- 
ows of opaque bodies consist of two parts, the umbra., or re- 
gion of no illumination, and i\iQ penumbra., or partial shadow. 

5. The intensity of illumination is inversely proportional 
to the square of the distance from a source of light (§ 280). 

6. The illuminating power of a source of light is meas- 
ured in terms of the candle power. This unit is about 
equal to the power of the ordinary household candle (§ 281). 



CHAPTER XIV 
LIGHT: REFLECTION AND REFRACTION 
1. REFLECTION OF LIGHT 

282. Reflection and Transmission. — It is a familiar fact 
tliat a piece of glass both reliects and transmits light ; for 
we frequently see the bright sunlight reflected by the 
glass of a window when we are outside, although, as we 
know, a large portion of the light is transmitted to the 
interior of the house. 

By means of a momited mirror M, Fig. 203, reflect a bright beam of 
sunlight upon a pane of glass AB, held obliquely. If a sheet of paper 

be placed behind the glass 
at t\ the transmitted light 
will fall upon it ; if, again, 
it be placed in the position 
A it will be brighfly il- 
luminated by reflected 
lisht. 








The ordinary mirror 
Fig. 203. — Reflection and Transmission of makes USe of the re- 
Light by a Pane of Glass, fl^ 4.- £ ^^ ^ ^ J- 

flection ol light from 
the surface of the opaque film of mercury that covers its 
back. Polished metals are often excellent reflectors. 

283. The Law of Reflection. — Every one is accustomed 
to the manner in which light is reflected by a mirror on 
account of the many purposes which it serves in everyday 
life ; but the following experiments may be performed in 
order to establish the law which ordinary observation does 
not reveal : 



278 



LIGHT: REFLECTION AND REFRACTION 279 





Fig. 204. — Illustrating the 
Reflection of Light by a 
Mirror. 



^//M///M//M/m^/MMM//M/^/^^ 



1. By means of a mirror held in the hand, reflect a beam of sun- 
light in various directions, and observe the position of the mirror in 
each case. Attach a cardboard index so 
that it shall be perpendicular to the mir- 
ror, and observe how it points in relation 
to the beam of light before and after its 
reflection. It will be found that the in- 
dex always points in a direction midway 

between the 
direct and re- 
flected beams 
of light, as 
shown in Fig. 
204. 

2. Attach a block of wood to a plane 
mirror, and set it upon a line ruled across 

^,^^„^ a large sheet of paper. Place a small 

Fig. 205.-The Angleof Re- candle about a foot from the mirror at C, 
flection r Equals the Angle Fig. 205. Now place the eye near the plane 
of Incidence i. ^f ^^g paper, and set two pins in line with 

the image of the candle wick seen in the mirror. Draw a line, as BO, 
through these pins to the mirror. Draw also a line, as CO, from the 
center of the candle to the point where the first line intersects the 
mirror. Draw the line OS perpendicular to the mirror at this point. 
Angles COS and BOS will be found to be equal. 

Now part of the light from the candle Q follows line CO 
to the mirror and line OB after being reflected. Angle 
COS\^ called the angle of incidence, and angle BOS the 
angle of reflection. In every case it will be found that 
these two angles are equal. Hence, the angle of reflection 
equals the angle of incidence. It is to be observed that these 
two angles are in the same plane, which in Experiment 2 is 
represented by the plane of the paper. 

284. Diffused or Scattered Light. — Objects are visible 
to us either by the light which they emit, as in the case of 
the sun, a candle, or a live coal, or by the light which, 
after falling upon them from some luminous body, they 
scatter, or diffuse. Most objects, unlike a smooth piece of 



280 



A HIGH SCHOOL COURSE IN PHYSICS 



glass, reflect light in many directions. Thus the sunlight 
which falls upon the snow is diffused ; but when it falls 
upon smooth ice, it is reflected as from a mirror. This is 
because the tiny reflecting surfaces of snow lie in all con- 




(i) (2) 

Fig. 206. — Diffusion Compared with Reflection from a Smooth Surface. 



ceivable positions, as shown in (1), ¥ig. 206, while those 

of ice all lie in one smooth plane, as in (2). 

By the help of the light which objects send to our eyes, 

we judge of their distance, form, size, color, and brilliancy. 

Leaves, grass, flowers, etc., diffuse in every direction the 

sunlight that falls upon them. The moon also is visible 

because of the sunlight diffused from its illuminated sur- 
face ; and we are often able to 
trace the dim outline of the 
new moon, although it is in 
shadow, because of the sunlight 
which the earth diffuses back 
upon the moon's dark area. 

285. Image of a Point in a 

Plane Mirror. — It was found in 

§ 283 that light is reflected by 

^, a plane mirror so that the angle 

Fig. 207.— Production of an im- of reflection is equal to the 

age by a Plane Mirror. ^^-,^1^ ^f incidence. HeuCC the 

light which starts from the point A, Fig. 207, and 



/i; 



n ^-v 




LIGHT: REFLECTION AND REFRACTION 281 



A' 



. I ' ' 



y^r^. 



takes the direction AB is reflected by the mirror MN 
in the direction ^C, so that angle OBD equals angle ABB. 
All other rays that may be drawn from A to the mirror 
are reflected in the same manner ; and when the eye is 
placed at B or JE\ the reflected rays appear to come from 
a point A' behind the mirror. 

286. Waves and "Rays." — It is easy to conceive of a 
train of waves moving outward from the point A, Fig. 
208, and striking against a plane mirror MJ}^. The waves 
are sent back from the mirror as though they emanated 
from the point A' behind the mirror. Hence, to an eye 
placed at B the effect is just the same as though A' were 
the light-emitting point. It is obviously more convenient 
to locate the image of a point 
by the help of *'rays," as in 
Fig. 207, rather than by the use 
of waves, as in Fig. 208. How- 
ever, it should ahvays he remem- 
bered that a so-called " ray " of 
light is simply a symbol used to 
represent the direction taken by 
a portion of a wave. Thus that 
part of a wave of light which 
starts from A toward B in Fig. 

207 follows the course ABO. 

287. Image of an Object in a 

Plane Mirror. — By applying 

the law of reflection it is easy to 

H k locate the image produced by a 

^ plane mirror. Let AB^ Fig. 

Fig. 209. — Manner of Locating 209, be an object and MJSf the 

an Image Illustrated. mirror. Let AC he an incident 

ray from A drawn perpendicular to the mirror. The re- 
flected ray will take the direction OA. (Why ?) Let AB 




Fig. 208. — Reflected Waves of 
Light from A Seem to Come to 
the Eye from A'. 



A'"" 




■<-> 



282 



A HIGH SCHOOL COURSE IN PHYSICS 



be another incident ray from A, whose direction DU, 
after being reflected, is found by making angle EDF equal 
angle ADF, The image of A lies at the intersection A^ of 
the reflected rays CA and BE produced backward behind 
the mirror. It is plain from the equality of the triangles 
AOD and A' CD that AQ equals A'0\ i.e. the image of a 
point is as far behind a plane mirror as the point itself is 
in front. We may now employ this fact in locating the 
image of the point B at B^. 

288. Seeing the Image. — Imagine an eye to be placed 
at B, Fig. 209. Light enters the eye from the mirror 
MJV as though it came from^', although it actually comes 
from A. Similarly, the eye receives light by other rays 
as if its origin were at B', whereas it is really at B, 
There is nothing behind the mirror that concerns our 
vision, and the light is not propagated by the medium 
except in front of the mirror. The image is called a vir- 
tual image to distinguish it from the real images formed by 
small apertures (§ 279) and in other cases, to be studied 
later. 




Fig. 210. — a Result of Double 
Reflection. 



Fig. 211. — Diagram Illustrating 
Double Reflection. 



289. Double Reflection. — If two plane mirrors are placed 
at right angles to each other, as shown in Figs. 210 and 



LIGHT: REFLECTION AND REFRACTION 283 

211, it is clear that a large portion of the light emanating 
from a point A will be reflected twice, once at the surface 
of each mirror. Thus the ray AB is reflected from B to 
by the vertical mirror Oiltf, and from toward I) by the 
horizontal mirror OiV. Likewise, the ray AU takes the 
course AUFCr, being reflected at U and F. Now F€r and 
(7i>, and all other rays that have undergone double re- 
flection, diverge as though they emanated from the point 
A"\ which is at one corner of the rectangle A' A A" A'". 
It is therefore evident that three images may be seen by 
placing the eye in such a position as to receive light that 
has suffered two reflections as well as that which has been 
reflected but once. 

EXERCISES 

1. A pole is inclined at an angle of 45° to the surface of water in a 
quiet pond. Construct the image of the pole seen in the water. 

2. Look at your image in a mirror, and lift your right hand. 
Which hand of the image appears to be lifted? Is the image direct 
or reversed ? 

3. Set a candle or a tumbler on a horizontal mirror, and observe 
the position of the image. 

4. What would be the result of covering the mirror OM in Fig. 
211 ? How could the formation of image A '" be prevented without 
interfering with image A' ox A"! 

Suggestion. — Place an opaque screen so that no light can be re- 
flected twice. Show how this can be done. 

5. A man approaches a plane mirror with a velocity of 3 m. per 
second. How rapidly is he approaching his image ? 

6. Try to read a printed page by looking at its image in a mirror. 
Write your name backward on a sheet of paper, and then look at the 
image of the writing in a mirror. W^hat effect is produced by the 
mirror in each case ? 

7. Find by construction the shortest vertical mirror in which a 
man 6 ft. tall can see his entire image when standing erect. 

Suggestion. — Diagram the case, and then draw lines from the 
man's eye to the highest and lowest points of the image. Consider 
the length of the mirror employed between these two limits. 



284 A HIGH SCHOOL COURSE IN PHYSICS 

8. Two lines AB and BC make an angle of 60° with each other. 
Show how^ to ]3lace a mirror so that AB may represent an incident 
ray of which BC is the reflected ray. 

2. REFLECTION BY CURVED MIRRORS 

290. Spherical Mirrors. — A spherical mirror is a 

polished or silvered portion of the surface of a sphere. 

,^-''"~"~\^ If the side of the surface toward the 

/ \ center of the sphere is used to re- 

/ ¥1 flsct light, see Fig. 212, the mir- 

I 1^ ror IS concave; if the outer surface 

\ S is used, the mirror is convex. The 

\ y^ center of the sphere is the center 

oio"""'ir'T' * of curvature, and the radius OC of 

Fig. 212. — Section of a ' 

Spherical Mirror. the sphere is the radlus of curvature. 

The line of symmetry XO is called the principal axis. 

Let direct sunlight fall upon a concave mirror parallel to the prin- 
cipal axis, and hold a small card in front of the mirror to receive the 
reflected light. Move the card back and forth until the illuminated 
spot is as small as possible. Measure the distance from this spot to 
the mirror. If a piece of tissue paper be held at the spot where the 
reflected light is concentrated, it will probably take fire. If the air in 
front of the mirror be filled with crayon dust, the convergence of the 
reflected light is easily made visible. 

Figure 213 shows the effect produced when parallel rays 
of sunlight fall upon a concave mirror. At every point 
on the mirror light is re- 
flected according to the law 
of reflection (§ 283) ; but 
on account of the curvature 
of the mirror, each reflected 

ray from the beam of par- F^«- 2?3- A Beam of Parallel Rays 

^ IS Focused at the Point F. 

all el rays is sent through 

the point F^ which is therefore called the principal focus. 

When the energy thus concentrated at the principal focus 




LIGHT: REFLECTION AND REFRACTION 285 

falls upon paper, enough of the energy is transformed into 
heat to ignite it. The principal focus is located midway 
between the center of curvature and the mirror; i.e. OF 
equals one half OC. The distance OF is called the focal 
length of the mirror or the principal focal distance. 

291. Convex Mirrors. — Try to concentrate direct sunlight by 
employing a convex mirror in the same manner as the concave mirror. 
While sunlight is falling upon the mirror, look toward its convex 
surface through a piece of black glass. An exceedingly bright point 
will be seen located apparently behind the mirror. 

Figure 214 illustrates the manner in which a convex 
mirror reflects the parallel rays of sunlight. The light at 
every point follows the law of reflection ; but on account 
of the form of the surface, it 
diverges as though it came 
from the point F behind the 
mirror. Of course no heat 
will be produced at the point 

F., inasmuch as the light does Fig. 214. — Parallel Rays are Made 
not actually pass through it. Divergent by a Convex Mirror. 

Since F is only an apparent meeting point or focus of the 
reflected rays, it is called a virtual or unreal focus. The 

principal focus of a con- 
cave mirror, however, 
is a real focus. See 
£. S288. 





To locate the principal 
focus of a convex mirror, 
let a beam of sunlight pass 
Fig. 215.— Determining the Focal Length through a round hole in a 
of a Convex Mirror. ^.^^^ ^^ cardboard, as shown 

in Fig. 215. Around this aperture draw a circle whose radius is just 
twice that of the aperture. Let the beam fall upon a convex mirror, 
and then move the mirror back and forth until the reflected light 
just covers the larger circle. Triangles abc and coF are practically 



286 



A HIGH SCHOOL COURSE IN PHYSICS 




.A' 



Fig. 



216. — Image Formed by 
a Convex Mirror. 



equal. (Why ?) Hence the distance from the cardboard to the mir- 
ror oe is equal to oF, the focal length of the mirror. 

292. Images Formed by a Convex Mirror. — The student 
is probably familiar with the small image that is seen as 

one looks at the polished sur- 
face of a glass or metal ball 
or the convex side of the 
bowl of a spoon. The experi- 
ment may be made with a 
lighted candle, as shown in 
Fig. 216. The image will ap- 
pear to he behind the mirror 
and is always smaller than the 
object^ which in this case is 
the candle. Compare with Fig. 208. 

293. Constructing the Image. — To show diagrammati- 
cally how an image is produced by a convex mirror, let 
the arc MN, Fig. 217, 
whose center of curva- 
ture is at (7, represent. 
a convex mirror. Let 
the arrow AB be the 
object, and draw the 
ray AGr parallel to 
the principal axis OX. 
The reflected ray GrD 
will apparently come 
from ^ (§ 291), which is midway between and C. Let 
a second ray AH be drawn from A along a radius of the 
mirror. Since this ray falls perpendicularly upon the 
mirror, it will be reflected back along the same line HA. 
Now let the reflected rays CrB and HA be produced until 
they intersect at some point, as a behind the mirror. This 
locates the image of the point A. The image of the point 




'-.::> a 



Q\ ^;r-~_-,.c 



Fig. 217. — Locating the Image of AB by 
Construction. 



LIGHT: REFLECTION AND REFRACTION 287 



B may be located in the same manner at h. A line drawn 
from a to h represents, therefore, the image of the object 
AB. Is the light from A actually focused at a ? Is the 
image therefore real or virtual? Is it erect or inverted? 
Is it larger or sinaller than the object ? Could any rays 
other than those selected be employed ? How would you 
find the direction of any other reflected ray ? (See § 283.) 
294. Images Formed by a Concave Mirror. — The nature 
of the images produced when light from some object, as a 
candle, falls upon a concave mirror is readily shown by a 
series of experiments. Excellent results can be obtained 
by using a mirror whose radius of curvature is 20 inches 
or more. 

1. Let a lighted candle be placed before a concave mirror at a 
distance somewhat greater than the radius of curvature. Place a small 






Fig. 218. — Production of a Real Image by a Coucave Mirror. 



cardboard screen between the candle and the mirror, and move it 
back and forth until a good image of the candle appears upon it. The 
image will be found between the principal focus and center of curva- 
ture. 

This image differs greatly from that produced by a con- 
vex mirror in that it can be caught upon a screen. We 
are not obliged to look into the mirror to see the image, 
because the light that emanates from a point in the candle 
is actually reflected to a corresponding point on the screen. 
Such an image is a real image. The* experiment plainly 



288 A HIGH SCHOOL COURSE IN PHYSICS 

shows that when the object is beyond the center of curvature^ 
the image is between the center and the principal focus^ is 
real^ inverted^ and smaller than the object. 

2. Let the candle be placed at any point between the center of curva- 
ture and the principal focus and the image caught upon a screen. In 
this case the screen has to be placed beyond the center. (See Fig. 218.) 

Here we shall readily find that when the object is placed 
between the center and the principal focus^ the image is 
beyond the center., is real^ inverted^ a7id larger than the object. 

3. Let the candle be placed between the principal focus and the 
mirror. In this case, in order to locate the image, direct the eye 
toward the mirror in such a manner as to receive some of the 
reflected light. An erect image will be seen. 

When the object is between the principal focus and the 
mirror^ the image is behind the mirror^ is virtual^ erect^ and 
larger than the object. 

4. Project upon a screen the images of some distant object, — clouds, 
trees, buildings, etc. In all cases it will be found that the images are 
small and lie practically midway between the mirror and its center' of 
curvature. Parallel rays of sunlight are also focused at this point. 

The experiment shows that the image of an object at a 
great distance lies near the principal focus, is real, inverted, 
and smaller than the object itself. 

295. Construction of Images Formed by Concave Mirrors. 
— We have seen in § 293 how an image can be located 
by geometrical construction. The same method may be 
applied to the cases arising from the use of a concave 
mirror. 

Case I. — When the object is beyond the center of curva- 
ture. 

Let MN, Fig. 219, be a concave mirror whose center is C Let the 
object be AB. Locate first the principal focus F (§ 290). Now let a 
ray ^ G^ be drawn from the point A of the object parallel to the 
principal axis. This ray will be reflected through the point F. 



LIGHT: REFLECTION AND REFRACTION 289 



(Why ?) Let a second ray AD he drawn through the center of curva- 
ture C. Since this ray is perpendicular to the surface of the mirror at 




Fig. 219. — A Real Image of AB is Located at ab. 

D, the reflected ray takes the direction DC. The two reflected rays 
GF and DC obviously meet at the point a. Could other incident rays 
be drawn from A ? How could their direction be found after reflection ? 
Where would they meet the reflected rays already drawn ? Hence the 
image of A is at the point a. In a similar manner the image of the 
point B is located at b. Thus ab is the image of the object AB. 

When two points are so related that the image of one 
falls at the other, as A and a, or B and b, they are called 
conjugate foci (pronounced /o' si). 

Case II. — When the object is between the center and the 
principal focus. 

The conjugate foci of the points A and B are located by a method 
similar to that used in the preceding case. (See Fig. 220.) The two 




Fig. 220. — A Real Image of AB is Located at ah. 

cases should be compared, and the constructions actually made. A 
real, inverted, and magnified image is. found at ab. 
20 



290 



A HIGH SCHOOL COURSE IN PHYSICS 



Case III. — WheM the object is between the principal focus- 
and the mirror. 

As we undertake here to carry out the method of construction used 
in the preceding cases, we find that the reflected rays emanating from 




Fig. 221. — ^ A Virtual Image of AB is Formed at ah. 

the point A diverge after leaving the mirror. (See Fig. 221.) This 
fact shows at once that A can have no real focus. The image will be 
seen only by looking into the mirror. In such cases ths reflected rays 
DC and GF are to be produced until they meet at some point as a 
behind the mirror. Similarly the image of B is found at h. Thus 
a magnified, erect, and cirtual image is found at ah. 

Case IV. — When the object is at the center of curvature or 
at the principal focus. 

When light from a point at the center of curvature falls upon a 
concave mirror, it strikes at an angle of 90° with the reflecting sur- 
face and is, consequently, reflected back along the same path. Hence 
all such rays will be focused at the center of curvature. But, when 
light from a point placed at the principal focus is reflected, it follows 
lines parallel to the principal axis ; e.g. FGA and FHB, Fig. 219. 
Since such rays never meet, no image of the point F could be 
produced. 

296. A Real Image Viewed Without a Screen. — We have 
already seen (§ 294) that a real image of a bright object can 
be projected by a concave mirror upon a suitable screen. 
Now if the eye be placed about 10 inches beyond the image 
and turned toward the mirror, the screen may be removed, 



LIGHT: REFLECTION AND REFRACTION 



291 





and the image will be visible. Imagine an eye at E^ 
Fig. 222. Light waves emanating from A^ a point in the 
object, advance toward the mirror with convex wave fronts, 
as those of water waves 
which are started by a 
falling pebble. These 

are here represented by [ j ^^nMJ 

arcs, having their com- ^kmMmmmmP^^^~ 
mon center at A. But 
on account of the curva- 
ture of the mirror MN^ 
the waves are reflected 
with concave fronts 
having as their common 
center the point a, which is called the conjugate focus of A. 
As the waves are not- obstructed by a screen, they leave a 
with convex fronts and enter the eye at E. 



222. — The Eye Can Observe a Real 
Image without a Screen. 



EXERCISES 

1. An object is placed 6 in. in front of a convex mirror whose 
radius of curvature is 12 in. Find by construction the position of 
the image. 

Suggestion. — Make a drawing, using for blackboard work the di- 
mensions and distances given, but divide each by 4 for pencil drawings. 

2. An object is placed 44 cm. from a concave mirror whose radius 
of curvature is 50 cm. Find by construction the location of the image. 

3. Place a small object slightly above the center of curvature of a 
concave mirror whose principal axis is horizontal, and find its image 
by construction. Does this exercise suggest a method for finding the 
radius of curvature by experiment ? 

4. An object 8 in. in height is placed 30 in. in front of a concave 
mirror whose radius of curvature is 15 in. Find the distance from 
the mirror to the image. 

5. Make an accurate construction of the case described in Exer. 4, 
and carefully measure the size of the image. Measure also the dis- 
tances of the object and the image from the center of curvature. Do 



292 A HIGH SCHOOL COURSE IN PHYSICS 

you find any relation between the distances and the sizes of image and 
object? If so, express it in a single sentence. Can you prove the 
same relation by similar triangles? 

3. REFRACTION OF LIGHT 

297. Refraction of Light. — Numerous examples of the 
refraction or bending of the course taken by light come 
before our attention daily, although we seldom give the 
phenomenon much thought. A simple case is the apparent 
bending of a spoon standing in a tumbler of water, or an 
oar at the point where it enters the water. Again, if a 
coin be placed in a tumbler of water and viewed obliquely, 
two coins become visible, — a small one seen through the 
horizontal surface of the water and a magnified one seen 
through the side of the vessel. If, now, a pencil be placed 
obliquely in the water contained in the tumbler, a bent 
section may be seen below the upper surface of the liquid, 
while a magnified portion is visible through the side of the 
vessel. In every case the illusion is due to the bending of 
light rays as they pass from one medium into another. No 
principles of optics are of more value to us than those relat- 
ing to the phenomenon of refraction, for upon this effect 

///, are based not only our 
///// most important optical 
Affcjiii^^^jii^^^^^^^^^^ instruments, including 
^ ^'%^^ \^ ^^ the microscope, tele- 

scope, and camera, but 
also the structure of 
the eye. 

298. Refraction II- 

FiG. 223. — Refraction of Light as it lusttated. — 1. By means 

Enters Water. „ . , . ,.. „. ^^^ 

of the mirror M, Fig. 223, 

about one half an inch in width, let a beam of sunlight be re- 
flected obliquely upon the surface of water in a tank. By scat- 
tering crayon dust in the air above the water, the course of the 



LIGHT: REFLECTION AND REFRACTION 293 



beam before and after entering the water becomes visible. Another 
excellent way to make the path of the light easy to trace is to hold 
a piece of white cardboard or tin partly under water so that it receives 
the beam of light both above and below the liquid surface. The result 
will show that at the surface of the liquid the beam of light turns 
toward the perpendicular, or normal, NN' which is drawn at 0. If 
the obliquity of the incident light is increased, the bending of the 
beam is made more pronounced. 

This experiment shows that when light passes obliquely 
from air into water, it undergoes a refraction or bending 
toward the perpendicular to the surface at the point where the 
beam enters the water. 

2. Place a plane mirror in the bottom of the tank used in the pre- 
ceding experiment so as to reflect a beam of light from water into the 
air, as shown in Fig. 224. The course 
of the beam MOP may be traced before 
and after entering the air by employing 
the means used in the preceding experi- 
ment. In fact, the path of the light 
MOP is precisely the reverse of that 
which the beam would take if it were 
passing into water along the line PO. 




Fig. 224. — Refraction of 
Light as it Emerges from 
Water. 



It will readily be observed in 
this experiment that when light 
passes obliquely from water into air it undergoes refrac- 
tion away from the perpendicular to the surface at the point 
where the beam emerges from the water. 

The angle MON, Fig. 223, is called the angle of inci- 
dence, and angle P ON', the angle of refraction. The per- 
pendicular OiVis usually called the normal at the point 0. 

299. Cause of Refraction. — It has been found by direct 
experimentation that light waves travel with less speed in 
water than in air ; in fact, the speed of light in water is 
almost exactly three fourths of that in air. When a beam 
of light AB, (1) Fig. 225, strikes at right angles to the 
surface of water QB, all parts of a given wave front strike 



294 



A HIGH SCHOOL COURSE IN PHYSICS 



A 








1 KS5^i 








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' \ 


■ \ 


■ 1 


\/\X.^ 










vvhX^ 










^'^ XXbKV 


c- 








\!/\/\/\ 








cxk \^y\^ 










l\^\^^ 










Water \ \>\^A^ 










|o'\>V^ 












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■ ' 


' > 


f ! >CAC 










1 \^^.<S( 




















N'\ y*, 










b 








fVi 



'd' 



the medium at the same time. Within the water the 
waves travel with less speed and are shorter. Likewise, if 
all parts of the wave front emerge at the same time, they 
resume their original speed without being refracted. But 
when the waves fall obliquely upon the surface, the case is 

quite different. See 
(2), Fig. 225. The 
parts of a given wave 
front ahcd do not enter 
the medium at the 
same instant ; but a 
enters first and con- 
, , tinues with reduced 

U) (2) 

Fiox. 225.— Illustrating the Cause Speed, while the other 

of Refraction. parts bed are still in air. 

Similarl}^ b enters the medium before c and c?, then c be- 
fore c?, and finally d. Thus the portion a travels the dis- 
tance aa', while d is traveling the larger distance dd' . 
The result is that the wave is " faced " in a different di- 
rection, namely aF, having suffered a bending toward the 
normal to the surface, I^W. From this explanation of re- 
fraction it is clear that the ratio of the distance dd' to the 
distance aa' is the same as the ratio of the speeds of light in 
the two media. 

300. Index of Refraction. — It is obvious from Fig. 223 
that the amount which the course of light is changed when 
it enters a medium where its speed is less than it is in air 
depends upon the relation that the distance aa' bears to 
the distance dd'; or, in other words, upon the relation 
between the speeds of light in the two media. Although 
it is not easy to measure the speed of light in a medium, 
it is comparatively a simple matter to measure the amount 
of refraction and from this to compute the relative speed 
of light. The number which expresses the ratio of the speed 



LIGHT: REFLECTION AND REFRACTION 295 



of light in air to its speed in another medium is called the 
index of refraction of that medium. Hence 

speed in air dd/ 

aa' 



Index of ref raction^ = 



(1) 



speed in other medium 

301. Index of Refraction Measured. — By referring to 

Fig. 225 in which xx' is the normal at the point c?^ we 

observe that angle dad' — angle dd'x^ which is the angle 

of incidence. (Why?) Angle ad' a' = angle x'd't^ the 

dd' . 
angle of refraction. (Why ?) Now the ratio — - is de- 

ad 
fined in mathematics as the sine of angle dad\ and is writ- 
ten sin dad'. Similarly, the ratio -— is the sine of angle 



ad' a' ., and is written sin ad'a' . 

dd' 
sin dd'x 



Hence 



sin x'dT 



sin dad' ad' dd . , , ^ 

— TT— i = — ; = — , = index of refraction. (2) 

sin ad'a aa; aa ^ 

ad' 



Therefore the index of refraction of 
a substance is equal to the quotient ob- 
tained by dividing the sine of the angle 
of incidence by the sine of the angle of 
refraction. 

302. Index of Refraction by Experi- 
ment. — Let a glass cube ABCD, Fig. 226, be 
placed against a pin P set upright in a hori- 
zontal board or table. Place the eye at some 
point as E, and set two other pins F and G in 

line with P as seen through the glass. Draw „ ^^^ ^^ . ^, 

° ^ Fig. 226. — Measuring the 

line AB and remove the cube. Next draw ludex of Refraction of 
lines GFo and oP. The line PoG is the Glass. 

i The term " absolute " index is used to refer to the ratio of the speed 
of light in a vacuum to its speed in a medium. The index of refraction 
defined above is often called the " relative " index. 




296 A HIGH SCHOOL COURSE IN PHYSICS 

course taken by the light that enters the eye from the pin P. Draw 
the normal hd at the point o, and then draw the lines ah and cd per- 
pendicular to the normal after making oa = oc. Now ab -^ ao is the 
sine of angle aob, and cd -^ co is the sine of angle cod. Hence, by 
equation (2), and substituting ao for its equal, co, we have 

ab 

. , /./.,. ao ab 

index of refraction = — - = —. 
. -^ cd cd 

ao 

Therefore we can readily find the value of the index of refraction by 
dividing the length ab by the length cd. 

Relative Speed of Light or Absolute Indexes of Refraction 
FOR Some Common Transparent Substances 

Air 1.00029 Flint glass .... 1.54 to 1.71 

Water 1.333 Carbon disulphide . . . 1.61 

Crown glass .... 1.51 Diamond 2.47 

303. Total Reflection. — Since the rays of light which 
pass from water or glass into air are bent away from the 
^^ normal, as PDA and PO'B, Fig. 

B 227, it is readily observed that 
Air Vo^^-^o^^^^^ s ^^^^^ ^^® angle of incidence below 
Water /// J^^\ ^^c surfacc is great enough, the 

refracted ray will follow close to 
the surface, as ray PO"S. Hence 
^ the ray P 0" is the last one that 

Fig. 227. — The Total Reflec- can emerge from the surface. If 
^^^ ° ^s • the angle of incidence is still in- 

creased, the light is reflected wholly beneath the surface, 
as PC' Q. This phenomenon is called total reflection. 

1. Examine the glass cube used in § 302 by looking through the 
face AB, Fig. 226, toward face BC, which has the appearance of a 
mirror. Place the finger upon face BC. It cannot be seen through 
AB. Transfer the finger to face CD. It can now be seen in face BC 
by rejfiected light. 




LIGHT: REFLECTION AND REFRACTION 297 



2. Look obliquely upward against the surface of water in a tum- 
bler. It will be seen to have the appearance of a plane mirror. If 
the point of a pencil is held in the water, only that part of it is visible 
that projects below the surface, and this portion can also be seen by 
reflected light. 

3. Reflect a narrow beam of sunlight obliquely upward through 
the side of a glass tank containing water. (See Fig. 228.) By vary- 
ing the angle of incidence below the 
surface until it is greater than 4:8.5°, 
the totally reflected beam can be traced 
back into the water. Both the incident 
and reflected beams may be made vis- 
ible in the manner described in 
§298. 

4. Hold a test-tube obliquely about 
5 centimeters under water, and look vertically downward upon it. 
The portion of the tube below the liquid surface has the appearance 
of a mirror. 




Fig. 228. — An Illustration of 
Total Reflection. 



a' 



These experiments serve to illustrate the fact that when 
the angle of incidence with which light undertakes to 
emerge from glass or water exceeds a certain value, the 
light is totally reflected at the point of incidence hack into the 
medium. The angle of incidejice at which the effect changes 
from refraction to total reflection is called the critical angle. 

The phenomenon of total re- 
flection occurs only when light 
is proceeding in one medium 
toward another in which the 
speed is greater. (See Fig. 
229.) The critical angle for 
water is 48.5°; for crown glass, 
41°; for diamond, 24°. 

304. Critical Angle Con- 
structed. — If the index of re- 
fraction of a medium is known, the critical angle can be 
found readily from the following construction : 




Fig. 229. — Illustrating the 
Critical Angle. 



298 



A HIGH SCHOOL COURSE IN PHYSICS 



and let the iudex of refraction be f . 



Let the line AB, Fig. 230, be the boundary between air and water, 

With 0, the point of incidence, 
as the center draw two concentric cir- 
cles whose radii have the ratio of 4:3. 
Since the ray that emerges for the crit- 
ical angle follows the surface, erect the 
normal at the point C, where the sur- 
face intersects the inner circle. This 
cuts the larger circle at E. The line 
EO produced below the surface gives 
the angle DOF as the critical angle. 

The proof is as follows : When light 
passes from water into air, the index of 
refraction is | ; and for the critical angle 
of incidence below the surface, the angle of refraction is 90°. It is 

to be shown that = -. Now, angles DOF and OEC are 

sin 90° 4 ^ 

equal. (Why ?) Further, sin OSC = ^ = - by construction (§ 301), 

and sin 90° = 1. Therefore 




t 



Fig. 230. — Method of Con 
structing the Critical Angle. 



sin DOF _ sin OEC _ 3 
sin 90° ~ sin 90° ~ 4 



1 = §, 



m 



m 



305. Total Reflecting Prism. — The most important use that is 
made of the phenomenon of total reflection is accomplished by em- 
ploying a right-angled prism of glass whose cross 

section is an isosceles triangle, as shown in Fig. 
231. If incident light enters the prism perpendic- 
ular to either of the faces forming the right 
angle, it will not suffer refraction and will strike 
the oblique face at an angle of incidence of 45°. 
Since the critical angle for glass is less than 45°, 
the light will be totally reflected in the direction 
perpendicular to the third face, from which it 
will emerge without undergoing refraction. Such 
prisms are used when it is desired to turn the 
course of light through an angle of 90° without excessive loss. 

306. Path of Light through Plates and Prisms. — The 

effect of a parallel-sided plate of glass upon a ray of light 
is readily determined by the following experiment : 



Fig. 231. — Turning 
the Course of 
Light through 90° 
by Total Reflec- 
tion. 



LIGHT: REFLECTION AND REFRACTION 299 




Fig. 232. — Light Suffers no 
Permanent Change in Di- 
rection in Passing through 
Parallel Surfaces. 



Look obliquely through a glass cube or a parallel-sided tank of 
water and set four pins, two on each side of the cube or tank, so that 
they will form apparently a straight line. Remove the refracting ob- 
ject, and draw the lines connecting the two pins of each pair. If these 
two lines are produced, it will be found that they are parallel. 

Hence, when light passes obliquely through a medium 
with parallel faces, it does not suffer ^a permanent change 
in direction. In other words, the ^e 
refraction toward the normal at the 
first surface BO, Fig. 232, is can- 
celed by the refraction from the nor- 
mal at the second surface AB. As 
the experiment shows, the ray suf- 
fers a lateral displacement only. 

The course taken by a ray of light 
which falls obliquely upon one of 
the faces of a glass prism may be 
traced in a manner similar to that just described: 

Set four pins, two upon each of the opposite sides of a glass prism, 
Fig. 233, whose angle at A is about 60°, so that all four pins lie 

apparently in a straight 
line. Draw a line around 
the prism and remove it. 
Join the two pins in each 
pair by a straight line, and 
produce these lines until 
they meet the sides of the 
prism AB and AC dX 
and 0'. Then 00' is the 
path of the light through 
the prism. 

Since the ray which enters the prism is bent toward the 
normal at and away from the normal at 0' wliere it 
emerges, it is clear that the effect of the prism is to turn 
the ray always away from the refracting angle A of the 
prism. 




Fig. 



233. — Refraction of Light by Means 
of a Prism. 



300 A HIGH SCHOOL COURSE IN PHYSICS 

EXERCISES 

1. What is the speed of light in water, the index of refraction be- 
ing I? The speed of light in air is 186,000 mi. per sec. 

2. Compute the speed of light in crown glass, assuming that the 
index of refraction is ^. 

3. Compare the speed of light in water with that in crown glass. 
What simple fraction will represent the relative speed ? Show how to 
get this same fraction from the indexes of refraction given in Exer- 
cises 1 and 2. 

4. The answer obtained in Exer. 3 is called the relative index of 
refraction for light on passing from water into crown glass. In a 
similar manner find the relative index of refraction for light which 
passes from fiint glass into carbon disulphide, using the indexes given 
in the table. 

5. For a given angle of incidence will light be refracted more in 
passing from air into crown glass or from water into crown glass? 
From water into crown glass or from carbon disulphide into flint 
glass? 

6. Construct the critical angle for air and water. On which side 
of the boundary surface does the critical angle lie ? Does the critical 
angle lie within tlie medium where the speed of light is the greater 
or the less? 

7. Upon which side of the boundary surface separating water and 
crown glass does the critical angle lie? 

8. The angle of incidence at which a ray of light enters a medium 
from air is 45°, and the angle of refraction 38*^. Find by construction 
the index of refraction of the medium. 

Suggestion. — By the help of a protractor construct accurately 
a figure. Measure the lines ab and cd, as in § 302, and compute the 
index of refraction. 

9. Draw the figures, and ascertain the indexes of refraction for the 
following angles : angle of incidence 30^^, angle of refraction 22^\ angle 
of incidence 50^, angle of refraction 31^"* ; angle of incidence G0"\ angle 
of refraction 40°. 

4. LENSES AND IMAGES 

307. Lenses. — The student will call to mind many in- 
struments with which he is acquainted that make use of 
of lenses, — the camera, microscope, spyglass, spectacles, 



\ 



LIGHT: REFLECTION AND REFRACTION 301 



opera glass, etc. The value of these instruments can 
hardly be too highly estimated. The study of lenses 
and their application is therefore of great interest and 
utility. 

A lens is usually made of glass and has either two 
curved boundary surfaces or one curved and one plane 
surface. The curved surfaces are usually (not necessarily) 
spherical, and the lens thus formed is 
called a spherical lens. The center of 
the sphere (7, Fig. 234, of which the 
lens surface is a part, is called the 
center of curvature^ and the radius of 
the sphere, the radius of curvature. 

Lenses are of two general classes, 
— convex lenses, which are thicker at 
the middle than at the edge, and con- 
cave lenses, which are thickest at the edge. Figure 235 
shows the three lenses belonging to each of the two gen- 
eral classes. 




Fig. 234. — a Spherical 
Lens. 



/ 



^^ "s^ 




13 





!s 



,/ 



Fig. 235. — Forms of Lenses. 



Convex, or Converg- 
ing Lenses 



1. Double Convex 

2. Piano-Convex 

3. Concavo-Convex, 
or a Meniscus 




Concave, or Diverg- 
ing Lenses 



~X 



16 



4. Double Concave 

5. Piano-Concave 

6. Convexo-Concave 



308. Effect of a Convex Lens on Light. — The most im- 
portant feature of any lens is not its form, but the manner 



302 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 236. — Effect of a Convex Lens on 
Parallel Rays. 



in which it acts upon light. The effect of a lens that 
is most familiar to all is that employed in the so-called 
" burning glass," which was in general use for producing 
fire before the introduction of cheap matches. The 
following experiment shows this effect : 

Allow a beam of sunlight to fall upon a large convex lens in a 

darkened room. If the air 
be made dusty, the light 
will be seen to form a cone- 
shaped figure as A A, Fig. 
236. If a piece of tissue 
paper be placed at the ver- 
tex of the cone, it is readily 
ignited. Beyond this point 
the light diverges precisely 
as from the principal focus 
of a concave mirror (§ 290). 

If a convex lens of another form be substituted for the first, the action 

is practically the same. 

The vertex of the cone formed by the sunlight trans- 
mitted by a convex lens is called the principal focus of the 
lens. The distance from the principal focus to the center 
of the lens is the focal lengthy or principal focal distance^ 
of the lens. Since all convex lenses cause rays of sun- 
light to converge to a point, they are often called converging 
lenses. (See Fig. 235.) 

Figure 237 shows the 
change in form that 
waves of light undergo 
while passing through 
a double convex lens. 
The plane waves of 
sunlight, whose direc- 




FiG. 237. — Plane Waves are Converged to 
the Principal Focus F. 



tion of motion is represented by arrows drawn in the figure, 
are retarded by the lens in proportion to the thickness of 



LIGHT: REFLECTION AND REFRACTION 303 

the glass through which they pass. As a result of this re- 
tardation, the emerging light has concave wave fronts 
whose centers are at F^ the principal focus. On leaving F^ 
however, the waves have convex fronts ; or, in other words, 
the light diverges from the point F. 

Ordinary glass lenses, whose index of refraction is very 
nearly |^, have a focal length equal to the radius of curva- 
ture, if they are double-convex and the two surfaces have 
the same curvature. If one surface is a plane, the focal 
length is double the radius of curvature. 

309. Concave Lenses. — The effect of a concave lens on 
sunlight is very different from that of a convex lens, as 
the following experiment will show : 

Let a beam of sunlight fall upon a concave lens mounted in a wide 
board. By making the air dusty, or holding a piece of cardboard 
obliquely in the transmitted light, it will be readily observed that 
the light diverges as it leaves the lens. 



The part of a plane wave that passes through the center 
of a concave lens. Fig. 238, is retarded least on account of 
the fact that the lens is 
thin at that point, while 
that passing^ through 
near the edge suffers 
the greatest retarda- 
tion. The result is that 
the front of the emerg- 
ing waves is convex, as 
though the light had 
emanated from the point F, which in this case is a virtual 
focus and located very near the center of curvature. Since 
the general effect of all concave lenses is to cause rays of 
sunlight to diverge, they are classed together as diverging 
lenses. (See Fig. 235.) 




Fig. 238. —Parallel Rays of Light are 
Scattered by Concave Lenses. 



304 



A HIGH SCHOOL COURSE IN PHYSICS 



310. Conjugate Foci. — Place a candle flame close to a small 
hole P in a piece of tin >S', Fig. 239. Now set a convex lens about 
twice its focal length away from the hole P, and place a screen S' 
on the opposite side of the lens upon which will be produced a sharp 




Fig. 239. —The Conjugate Focus of P is at P'. 

image of P at P' . Cover up one half the surface of the lens, and the 
image will remain at P'. Cover the lens almost entirely, and the image 
will be weakened but not destroyed. 

It is to be observed that all the light emanating from 
the point P and passing through the lens is collected at 
P^ Likewise, any other point beyond the principal focus 
F has a corresponding point on the opposite side of the 
lens where its image would appear. Two points so related 
that the image of one of them, as P, falls at the other, as P' , 
are called conjugate foci. See also § 295. 

311. Virtual Focus. — Let the candle and screen ,S that were 
used in the experiment of the preceding section be placed between a 

convex lens and its principal 
focus, and the result shown 
in Fig. 240 will be secured. 
The rays diverging from the 
point P will not be brought 
to a focus by the lens, but 
the divergence will be greatly 
reduced. Of course no im- 
age of P can be produced 
upon a screen ; but upon looking through the lens the eye E locates 
(apparently) the image of P at the point P'. If the screen be removed 
a virtual image of the candle may be seen. 



P' 




Fig. 240: — The Conjugate Focus of P is 
Virtual and at Point P'. 



LIGHT: REFLECTION AND REFRACTION 305 



In the location of an image the eye is always governed 
by the divergence of rays which enter it (§ 276). The 
divergence is in this case as if the rays had come from P' 




Fig. 241. —A Concave Lens always Forms a Virtual Focus. 

instead of P. Since the point P' is only the apparent 
meeting place of the rays that enter the eye, this point is 
called the virtual focus of P. If the experiment be 
repeated with a diverging (concave) lens, as shown in 
Fig. 241, the conjugate focus of P is virtual no matter 
what the position of P may be, because the effect of such 
a lens is always to scatter the transmitted light. 

312. Images Formed by Convex Lenses. — Every one 
who has viewed the pictures projected by a stereopticon 
or a moving-picture machine has seen the real images that 
convex lenses are able to produce. In the process of 
forming images the convex lens presents precisely as 




Fig. 242. 



21 



Focusing a Real Image on a Screen by Means 
of a Convex Lens. 



306 A HIGH SCHOOL COURSE IN PHYSICS 

many cases as the concave mirror (§ 295). These cases 
are easily illustrated by experiments. 

1. Let a lighted candle C, Fig. 242, be placed at a little more than 
twice the focal length from a convex lens L. By moving the screen S 
back and forth an image will be found distinctly focused upon it. 
The distance from the lens to the image should be measured and 
compared with the focal distance. 

When the object is situated at more than twice the focal 
length from a convex lens, the image is at less than twice and 
more than once the focal length from the lens on the opposite 
side, is real, inverted, and smaller than the object, 

2. Let the candle be placed at less than twice but more than once 
the focal length from the convex lens. The image may be found by 
moving the screen away from the lens. The distance to the image 
should be measured and its position and size noted. 

When the object is situated at more than once and less 
than twice the focal length from a cofivex lens, the image is 
at more than twice the focal length on the other side, is real, 
inverted, and larger than the object. 

3. Let the candle be placed at less than the focal length from the 
convex lens. Of course no image can be produced upon the screen 
since the conjugate foci of all points in the object are virtual. (See 
§ 311.) But by allowing some of the transmitted light to enter the 
eye {i.e. by looking through the lens), an apparent magnified image 
may be seen behind the lens. 

When the object is situated at a point between a convex 
lens and its principal focus, the image is apparently behind 
the lens, is virtual, erect, and larger than the object, 

4. Focus upon a screen the images of objects which are situated 
at a great distance, — clouds, trees, buildings, etc. If, now, the dis- 
tance from the lens to the images be measured, it will be found prac- 
tically equal to the focal length of the lens. Repeat the experiment 
with different lenses. Let the size and position of these images be 
noted. 



LIGHT: REFLECTION AND REFRACTION 307 

When an object is at a great distance from a convex lens^ 
its image is at the focal distance from the lens^ is real^ 
inverted^ and smaller than the object. 

It will be observed that this follows from tlie fact that 
the rays which come from a given point on the object to 
the lens are practically parallel to each other like rays of 
sunlight. This case affords a good method for determin- 
ing the focal length of a lens. 

Since rays of light which diverge from the principal focus 
of a convex lens become parallel to the principal axis after 
passing through, it follows that there can be no image of a 
small object placed at the principal focus. 

313. Images Formed by Concave Lenses. — It is obvious 
from § 311 that a concave lens cannot produce a real 
image, since it always tends to scatter the light. This 
fact, however, is of value in the construction of certain 
optical instruments, as will appear later. 

Hold a concave lens between the eye and a candle flame. No mat- 
ter how far the flame is from the lens, the only image produced is a 
small erect one behind the lens. 

Hence the image produced by a concave leiis is always ap- 
parently on the same side of the lens as the object^ is virtual, 
erect, and smaller than the object. 

314. Construction of Images Formed by Lenses. — The 
construction of the images produced by lenses will serve 
to bring out clearly the nature of the image in each of the 
cases illustrated by experiment in § 312. 

The different cases of lenses now to be studied will be 
found to correspond closely to those of curved mirrors 
which were treated in section 295. In each case the com- 
plete construction should be accurately made on a scale 
somewhat larger than that used in the illustrations as 
here given. 



308 



A HIGH SCHOOL COURSE IN PHYSICS 



Case I. — When the object is placed at more than twice the 
focal distance from a convex lens. 

Let ED, Fig. 243, be a convex lens whose centers of curvature are 
at C and C. Let AB represent an object so placed that the distance 




Fig. 243. — Method of Locating the Image Produced by a Convex Lens. 

01 is more than twice the focal length OC. Now if rays AE and BD 
be drawn parallel to the principal axis HI, the refracted rays will 
pass through the principal focus i^, which is at C (§ 308). Again, 
tlie rays that pass through the optical center of the lens enter the 
lens and emerge from it at points where the surfaces are parallel and there- 
fore suffer no permanent change in direction. (See § 306.) Let the 
line AOhe, produced through the lens until it meets the line EFat a. 
Thus a is the conjugate focus of A. Similarly b is found to be the 
conjugate focus of B. Therefore ab is the image of the object AB. 

Case II. — When the object is at more than once and less 
than twice the focal distance from a convex lens. 

In this case the method of construction is precisely the same as in 
the preceding one; but on account of the fact that the object has 
been brought nearer the lens, the divergence of the rays before refrac- 

Ii6ii 




Fig. 244. — Illustrating the Production of a Keal and Magnified Image. 

tion (which is represented by angle OAE) is greater than before, and 
hence they are brought to a focus a at a greater distance from the lens. 



LIGHT: REFLECTION AND REFRACTION 309 

Figure 244 shows clearly the construction. Does the description given 
in § 312 apply to the image that is found by construction ? 

It is to be observed that Case I changes to Case II when 
the object is placed at twice the focal length from the lens. 
In this instance the image and object are of equal size and 
are equidistant from the lens. 

Case III. — When the object is at less than the focal dis- 
tance from a convex lens. 

Two rays are drawn from each of the points A and B, Fig. 245, 
precisely as in the two preceding cases. But since the angle of 
divergence EA is so large, the lens is not able to cause the rays to 
converge to a real focus ; i.e. the refracted rays EF and A never meet 
after leaving the lens. If, however, the refracted rays enter an eye, an 
apparent image of A will be seen at a, which is the intersection of the 



/f^r ^ 




Fig. 245. — The Production of a Virtual Image by a Couvex Lens. 

refracted rays when produced behind the lens. Since the effect is an 
illusion, the image is a virtual one. Does the description given in 
§ 312 apply to this image? 

Case IV. — When an object is viewed through a concave lens. 




Fig. 246. — Constructing the Image Produced by a Concave Lens. 



310 A HIGH SCHOOL COURSE IN PHYSICS 

Rays A and BO, Fig. 246, are drawn as in the preceding cases. 
But the parallel rays AE and BD are turned away from the principal 
axis CC as if their origin were at the center of curvature C (§ 309). 
Thus the angle of divergence EA is increased by the lens to the 
value EaO. Consequently, when the refracted light is received by an 
eye, the image which is virtual and erect appears to be behind the 
lens, but nearer and always smaller than the object. 

315. The Lens Equation. — The experiments of the preceding 
sections have shown that the position of an image formed by a lens is 
determined by the focal length of the lens and the distance from the 
lens to the object. If p represents the distance from the lens to the 
object, q the distance from the lens to the image, and / the focal 
length, then the following relation between these distances will be 
found to exist : 

i = Ui. (3) 

f p q V y 

Referring to Fig. 243, we observe that the triangles AOI and aOH 

are similar. Hence —7^ = 777^' If the lens be thin and a line be 
all On 

drawn from E to D, it may be assumed to pass through the point 0. 

EO OF 
Then the triangles EFO and aHFare similar. Therefore, -^7 = 777-,' 

Since Al = EO, the first member of these two proportions are equal. 
Hence, by substituting p for 01, q for OH, and / for OF, we obtain 

P^ f 

Clearing of fractions, transposing, and dividing by pqf gives 

f p q 

Equation (3) is useful in determining the focal length of a lens. 
For example, let the image of an object which is 60 cm. from a lens 
be focused 40 cm. from the lens. By substituting these values for p 
and q, we find the value of /, the focal length, to be 24 cm. 

316. Relative Size of Object and Image. — By referring 
to Fig. 248 and the above discussion the following propor- 
tion is found to be true: 

-^^^. (4) 

aH OH ^ ^ 



./*■ 



LIGHT: REFLECTION AND REFRACTION 311 

But alTis the image of AI. Therefore, the ratio of the 
size of the object to the size of the image is equal to the ratio 
of their respective distances from the lens. This relation 
may be easily verified by experiment. 

EXERCISES 

1. What kind of mirrors and lenses always produce virtual 
images? 

2. Under what conditions do convex lenses produce real images? 
When is the image produced by a convex lens virtual ? 

3. If one half of a convex lens be covered with an opaque card, 
what will be the effect upon the real images produced by it ? Test 
your answer by experiment. 

4. How can you test a spectacle lens to ascertain whether it is 
convex or concave ? 

5. By the help of equation (3) compute the focal length of a 
lens when the image of a candle flame 120 cm. away is focused at a 
distance of 60 cm. 

6. The focal length of a lens is 50 cm. If an object is situated 
at a distance of 75 cm. from the lens, how far from the lens will its 
image be focused ? 

7. An object is placed 30 cm. from a lens whose focal length is 
45 cm. Locate the image by employing equation (3). 

Suggestion. — When the minus sign precedes the result obtained 
by using the equation, the inference is that the image is virtual. 

8. What is the height of a tree 350 ft. away when its image on a 
screen 10 in. from a convex lens is 2 in. in height? Ans. 70 ft. 

5. OPTICAL INSTRUMENTS 

317. The Simple Magnifier or Reading Glass. — It is a 

common occurrence to see a botanist examining the details 
of a flower or a jeweler adjusting the minute parts of a watch 
by the help of a convex lens. Large convex lenses are 
frequently used as an aid in reading, and are therefore 
often called reading glasses. In these cases the object to 
be examined is placed a little nearer the lens than the 
principal focus, while the image is viewed by placing the 



312 



A HIGH SCHOOL COURSE IN PHYSICS 



eye on the opposite side of the lens, as shown in Fig. 240. 
The instrument owes its importance to the fact that a 
magnified image is visible behind the lens, as shown in 
Fig. 245. 

318. The Photographic Camera. — Two important prin- 
ciples are employed in the photographic camera : (1) a 
convex lens produces a real image of objects placed beyond 
its principal focus, and (2) light has the property of pro- 
ducing chemical changes in certain compounds of silver. 

The camera is a light-proof 
box, or chamber, Fig. 247, 
provided at the front with 
a convex lens L and at the 
back with a ground-glass 
screen which can be replaced 
by a "sensitized" plate or 
film for receiving the image. 
The image is first focused on 
the screen by varying its dis- 
tance from the lens, after 
which the sensitive plate is introduced in a light-proof 
holder. The shutter of the lens is now closed, and the 
cover of the plate-holder removed. When all is in readi- 
ness, the lens is uncovered for a sufficient time to enable 
the transmitted light to produce the desired effect upon the 
plate. The plate is now covered and taken to a dark room, 
where a "developing" process brings out a visible and per- 
manent image. The plate thus treated is called a " neg- 
ative " because of the reversal of light and shade in the 
picture upon it, and may be used in the reproduction of 
any number of positive photographs on prepared paper. 

319. The Eye. — Although the eye is a very complicated 
structure, its action depends on one of the simplest cases 
of refraction that we have studied. The eye is essentially 




Fig. 247.— The Camera. 



LIGHT: REFLECTION AND REFRACTION 313 




Fig. 248. — Sectional View 
of the Eye. 



a small camera at the front of which the cornea (7, Fig. 248, 

the aqueous humor A, and the crystalline lens take the 

place of the convex lens of that 

instrument. When the eye is 

directed toward an object, a 

small, real, and inverted image 

is produced on the retina, 

which is an expansion of the 

optic nerve iV and covers the 

inner surface of the eyeball at 

the back. This does not mean, 

however, that we see things 

upside down. The relative position which we ascribe to 

objects is the result of experience aided by the sense of 

touch, etc., and the fact that images are inverted on the 

retina has little effect on the ideas which the impression 

gives us. It will be observed that the impressions remain 

the same even when the eye is tilted or inverted. 

The eye adjusts itself to objects near and far by chang- 
ing the focal length of the crystalline lens. When a 
normal eye is completely relaxed, the lens has the proper 
curvature for focusing light from distant objects (i.e. par- 
allel rays) upon the retina. When, however, we wish to 
view an object near at hand, as in reading, small muscles 
within the eye cause the curvature of the crystalline lens 
to increase until the image is again focused upon the sen- 
sitive retina. 

320. Spectacles and Eyeglasses. — Although a normal 
eye with complete relaxation focuses parallel rays upon 
the retina, as in Fig. 249, it should also be able to focus 
with ease light that comes from an object at a distance of 
10 inches or more. The eye is defective when it cannot 
accomplish the performance of these functions without 
unnatural effort. 



314 



A HIGH SCHOOL COURSE IN PHYSICS 




when Relaxed. 




If the retina is too far from the crystalline lens, parallel 
rays will not be brought to a focus upon it, but in front 
of it. This defect is called myopia^ or near-sightedness. 

(i) Fig. 250 illustrates this con- 
dition. It is at once obvious that the 
crystalline lens produces too great a 
Fig. 249. — a Normal Eye convergence of the light. This can 

be corrected by using a concave lens 
of suitable curvature, as shown in 
(^) Fig. 250. Hence concave spec- 
tacles are used to assist myopic eyes. 
Again, the eyeball may be too short 
from front to back, in which case the 
focus of parallel rays will be behind 
the retina, as shown in (i). Fig. 251. 
It is clear in this instance that the 
crystalline lens does not converge the 
rays sufficiently for distinct vision. 
This defect is called hypermetropia^ or 
far-sightedness. In order to correct 
the fault, a 
convex lens 
of suitable 

Fig. 251. — The Hyperme- , 

tropic Eye and Its Cor- CUrvatUre 

rection. is needed, 

which assists the crystalline lens 
to produce an adequate conver- 
gence of the light, as shown in (2). 

The most prevalent defect in 
the eyes of young people is that 
known as astigmatism-. Astigma- 
tism is due to the fact that the crystalline lens is not sym- 
metrical about its axis. In other words, a vertical section 
through the lens differs in form from a horizontal section. 



Fig. 250. — The Myopic Eye 
and Its Correction. 



--(I) 





Fig. 252. — An Astigmatic Eye 
Sees these Lines with Unequal 
Distinctness. 



LIGHT: REFLECTION AND REFRACTION 315 

Such an eye sees the lines of Fig. 252 with unequal distinct- 
ness. This defect is corrected by the use of a lens whose 
vertical and horizontal sections possess suitable curvatures 
to make up for the deficiencies in the crystalline lens. 

321. The Compound Microscope. — The compound micro- 
scope consists of a convex lens 0, Fig. 253, of short (say Jin.) 
focal length, which is called 
the objective^ and a larger con- 
vex lens E^ called the eye- 
piece. When the object to 
be viewed, AB^ is placed a 
little beyond the principal 
focus of 0, a real, inverted, 
and magnified image is pro- 
duced at a5(§312). When 
this real image is viewed 
through the eye-piece, a 
magnified virtual image is 
seen at a'h' , 

322. The Astronomical 
Telescope. — The principal 
part of a modern astronomi- 
cal telescope is the large 
convex lens 0, Fig." 254, 




^a' 



Fig. 253. — The Compound Microscope. 



called the object glass. This is designed to collect a large 
amount of light in order that the real inverted image ah 
that is formed may be sufficiently brilliant. This image 




Fig. 254. — The Astronomical Telescope. 



316 



A HIGH SCHOOL COURSE IN PHYSICS 



falls close to the eye-piece E whose function is precisely 
the same as in the compound microscope which is described 
in the preceding section. The inversion of the image is of 
little consequence in astronomical telescopes, but for view- 
ing terrestrial objects this feature would be a defect. 

323. The Opera Glass, or Galileo's Telescope. — The 
honor of having invented the original form of the tele- 




FiG. 255. — Illustrating the Principle of the Opera Glass. 

scope belongs to Galileo,^ who constructed the first instru- 
ment about 1610. Two Galilean telescopes arranged side 
by side form an opera glass or a field glass. An object 
glass 0, Fig. 255, converges the light to form a real image 
at ah. Before the rays reach the focus, however, they are 
intercepted by a concave lens which gives them a slight 
divergence as they enter the eye. As a consequence, an 
erect and magnified virtual image is seen at a'h' . 

324. The Projecting Lantern. — The projecting lantern, 
Fig. 256, consists of a powerful source of light A whose 




Fig. 256. — Illustrating the Principle of the Projecting Lantern. 
1 See portrait facing page 70. 



LIGHT: REFLECTION AND REFRACTION 317 

rays are concentrated upon a transparent picture B by the 
convex condensing lenses L. At the front of the instru- 
ment is placed the projecting lens P which forms a real, 
inverted, and enlarged image of B upon the screen S, 
The source of light is usually an electric arc lamp or 
a calcium light. The latter is produced by directing an 
exceedingly hot flame, produced by burning a mixture of 
hydrogen and oxygen, against a piece of lime. When 
raised to a high temperature, the lime becomes intensely 
luminous. 

325. Binocular Vision. — The Stereoscope. — On account of 

the fact that the two eyes are separated by a distance of 6 or 7 centi- 
meters, the images produced upon the two retinas are not precisely alike. 
This has the effect of giving to an object the appearance of solidity or 
depth. Advantage has been taken of this fact in the stereoscope, an 
instrument which is so constructed as to present to each eye a similar 
image to that w^hich it would receive if the object itself were present. 
A double photograph is first made by means of a camera having two 
objectives separated by a distance about equal to that between the 
two eyes. The two pictures thus taken differ just as much as would 
the corresponding images upon the 
retinas of the eyes. These pictures 

are now mounted on the stereoscope d ^^^^ ----'Ik 

at A and B, Fig. 257, so as to be A^ "^' i i 

viewed by the two eyes through the f ^ " 

half -lenses m and n. These lenses Fig. 257. — Illustrating the Principle 
are so adjusted that the images pro- ^^ *'^® Stereoscope. 

duced upon the retinas are related in position precisely as in ordinary 
vision. On this account a perfectly natural blending of the two im- 
pressions is brought about as though the object itself were in the di- 
rection of C. The observer is therefore conscious of the presence of 
only one photograph, which gives the effect of extension in a degree 
that is remarkably true to nature. 

326. The KinetOSCOpe The kinetoscope, kinematograph, or 

moving-picture machine, is a common object in most cities and 
villages. A life-like motion is given to pictures projected on a 
screen by means of a series of transparent photographs taken as 
follows : 



318 A HIGH SCHOOL COURSE IN PHYSICS 

A camera is provided with a shutter that opens and closes auto- 
matically about 12 times a second. The instrument also contains a 
long, narrow, sensitive film which moves along about 2 centimeters while 
the shutter is closed, and remains stationary while the shutter is open. 
With this a series of pictures is taken, each of which differs slightly 
from the preceding, proiu'rfec^ any moving object is in the field of the camera. 

These pictures are thrown upon a screen with a projection lantern 
in precisely the same order and with the same rapidity as they were 
taken. On account of the fact that the sensation produced by one 
picture remains until the next picture appears, the observer is uncon- 
scious of any interruption in the illumination of the screen upon 
which the pictures are produced. 

EXERCISES 

1. While changing the attention from a distant object to a ne?T 
one, does the crystalline lens flatten or thicken ? 

2. A photographer finds that the desired image of a building more 
than covers the area of the plate to be used. How can the size of the 
image be reduced to fit the plate ? 

3. When the spectacle lens used to correct a myopic eye is placed 
in front of a normal eye, is the image of a distant object behind or in 
front of the retina? 

/ 4. How can you test your eyes for astigmatism ? 

5. The crystalline lens becomes less elastic with age. Account for 
the spectacles with double lenses which are frequently worn. 

6. The picture projected on a screen by a projecting lantern is 
found to be too large. Which way must the instrument be moved in 
order to reduce its size? 

7. Why is it necessary to " focus " a microscope or a telescope upon 
the object to be viewed? 

SUMMARY 

1. Light is reflected from polished surfaces in snch a 
manner that the angle of reflection equals the angle of 
incidence (§§ 282 and 283). 

2. Light is diffused from unpolished surfaces. It is 
the power of diffusing light that renders objects visible 
from different points of observation when light falls upon 
them (§ 284). 



LIGHT: REFLECTION AND REFRACTION 319 

3. The image produced by a plane mirror is as far be- 
hind the mirror as the object is in front of it (§§ 285 to 289). 

4. The tendency of a concave mirror is to collect rays of , 
light. Thus parallel rays are reflected through a common 
point called ih.Q principal focus (§ 2902). 

5. The tendency of convex mirrors is to scatter light. 
Hence the principal focus is unreal, or virtual (§ 291). 

6. The images formed by a convex mirror are always 
virtual, erect, smaller than the object, and behind the 
mirror (§ 292). 

7. The images formed by a concave mirror depend on 
the position of the object relative to the center of curva- 
ture (§ 294). 

8. Light is refracted or bent toward the perpendicu- 
lar to the surface where it enters a medium — as glass or 
water — from the air, and away from the perpendicular 
when it emerges from the medium into the air. Refrac- 
tion is due to the fact that the speed of light is less in 
water, glass, etc., than in air or a vacuum (§§ 297 to 299). 

9. The index of refraction of a medium expresses the 
ratio of the speed of light in air to its speed in that 
medium (§§ 300 to 302). 

10. Total reflection always takes place when light 
travels in one medium toward another in which its speed 
is greater, provided the angle of incidence upon the 
boundary surface is greater than the critical angle (§§ 303 
to 305). 

11. Lenses are classed as convex or concave according to 
form, and as converging or diverging according to their 
effect on light (§ 307). 

12. Convex^ or converging, lenses tend to collect light. 
Rays which are parallel before reflection are caused to 



320 A HIGH SCHOOL COURSE IN PHYSICS 

pass through a common point, called the principal focus 
(§ 308). 

13. Concave^ or diverging, lenses tend to scatter light, 
and therefore the principal focus is unreal or virtual (§ 309). 

14. The images formed by convex lenses depend upon 
the relative position of the object and the principal focus 
(§ 312). 

15. The images formed by a concave lens are always 
virtual, erect, smaller than the object, and on the same side 
of the lens as the object (§ 313). 

16. Conjugate focal distances p and q are related mathe- 
matically to the focal distance / as shown by the following 
equation : 

l = K\ (§816.) 

17. The sizes of image and object are in proportion to 
their respective distances from the lens (§ 316). 

18. The convex lens is employed in the reading glass, 
camera, spectacles, and the eye. Two or more convex 
lenses are used in the compound microscope, telescope, 
projecting lantern, etc. Concave lenses are used in some 
spectacles, and in the eye-piece of the opera glass (§§ 317 
to 326). 



CHAPTER XV 



LIGHT: COLOR AND SPECTRA 



1. DISPERSION OF LIGHT: COLOR 
327. Decomposition of White Light. — Let sunlight pass 

through a narrow slit into a well-darkened room and fall obliquely on 

a glass prism, as shown 

in Fig. 258. As previ- \Sunlight^ 

ously shown (§ 306), the 

beam will be refracted; 

but when the refracted 

light is allowed to fall on 

a white screen, a beautiful 

band of different colors 

will be seen. Among 

these will be recognized 

red, orange, yellow, green. 




Fig. 258. — The Separation of White Light into 
its Components. 



blue, indigo, and violet, although there is no sharp line of demarca- 
tion between them. 

From this experiment we can only infer that white light 
is a mixture of the several colors seen on the screen. In 
fact, if a similar prism is available, the colors may be 

turned together again to form 
white light by placing the two 
prisms as shown in Fig. 259. 
The band of color produced 
by the separation of light of 
any kind is called a spectrum, 
and the process of separation 
in which the light of different colors is refracted in vary- 
ing degree is called dispersion. 
22 321 




Fig. 259. — Colors of the Spectrum 
Added to Produce White. 



322 A HIGH SCHOOL COURSE IN PHYSICS 

328. Cause of Dispersion. — Direct measurements of the 
waves of light of different colors show that they are of very 
different lengths, those of red light being longest and of 
violet shortest. It is therefore clear that waves of differ- 
ent lengths are refracted unequally by the prism, — the red 
or longest waves being bent least, and the violet or shortest 
waves most. The following table shows the approximate 
wave lengths of the various colors: 

Wave Lengths of Light 



Red . . 


. 0.000068 cm. 


Green . . 


. 0.000052 cm. 


Orange . 


. 0.000065 cm. 


Blue . . 


. 0.000046 cm. 


Yellow . 


. 0.000058 cm. 


Violet . . 


. 0.000040 cm. 




329. Achromatic Lenses. — When white light passes through a 
simple lens, it is dispersed as well as refracted, i.e. the violet light is 
brought to a focus somewhat nearer the lens than the red, since it suffers 
the greater refraction (§ 328). On this account the images formed by 
simple lenses are always fringed with color, which is a serious fault in 

optical instruments. It is, however, possible to 
remedy this defect by combining two lenses, — 
one being a double convex lens of crown glass, the 
other a plano-concave lens (plane on one side, 
Fig. 260. — Achro- concave on the other) of flint glass, as shown in 
matic Lens. Fjg. 260. In this lens system the dispersion pro- 

duced in one part is just neutralized by the other, 
while the refraction is reduced only about one half. Such a system 
of lenses is called an achromatic lens, since all color is eliminated. 

330. Color of Objects. — Let the spectrum of sunlight be pro- 
jected on a white screen. Hold small pieces of paper or cloth of differ- 
ent colors in the various parts of the spectrum, and observe the effect. 
A red object will be a brilliant red when held in the red of the spectrum, 
and black in other parts. Blue and violet objects show a coloration 
only when placed near the violet end of the spectrum. A black object 
will appear black everywhere, and a white one reflects that color of 
the spectrum in which it is placed. 

The experiment shows that the color of an object 
depends (1) on the light which falls upon it and (2) on 



LIGHT: COLOR AND SPECTRA 323 

the light it reflects to the eye. A red object is red be- 
cause it reflects mainly red light, all other incident light 
being absorbed by the material, i.e. transformed into heat 
(§ 259). If, therefore, no red light falls upon it, it can 
reflect no light and is black. A black object appears black, 
because it absorbs practically all colors alike ; and a white 
body is white because it reflects all colors to the same 
extent. 

331. Color of Transparent Bodies. —Project the spectrum of 

sunlight, and hold pieces of glass of different colors in the beam at 
any point. It will be observed that a red glass, for example, transmits 
mainly red light and absorbs the rest, that green glass transmits 
mainly green light, etc. Placing two pieces at once in the path of the 
light results in the transmission of that light only which neither can 
absorb. 

It is therefore clear that the color of a transparent body 
depends on the color of the light which it transmits. The 
color that we see in any body is the coynhination of all the 
light that is not absorbed by that body. A colorless body is 
one that transmits all colors equally. 

332. Complementary Colors. — Project the spectrum of sunlight 
or the light from an electric arc, 

and reunite the colors by means of 
a similar prism, as shown in Fig. 
261. The result is, of course, 
white. Now place a card C be- 
tween the prisms, and let it in- ^ ^ , ^ ^ 

, ,, -, T 1 , , rr^i Fig. 261. — Production of Coniple- 

tercept the red light only. The mentary Colors. 

spot of light on the screen S turns 

to bluish green. If the card is now caused to cut out the violet and 

blue, a greenish yellow results. 

Two colors, as red and bluish green, into which white 
light can be resolved are called complementary colors. The 
union of complementary colors results in the production of 
white. When any color, or combination of colors, is 




324 A HIGH SCHOOL COURSE IN PHYSICS 

removed from the spectrum of white light, the color pro- 
duced by combining the remaining colors is the comple- 
mentary of the part removed. The two complementary 
colors most easily obtainable are yellow and blue. If one 
half of a circular disk be colored blue and the other half 
yellow, the color effects may be combined by rotating the 
disk on a whirling machine. If the whirling disk be 
strongly illuminated, it appears white. 

Table of Complementary Colors 

Red and bluish green. Green and purple. 

Orange and greenish blue. Violet and greenish yellow. 

Yellow and blue. 

333. Color of Pigments. — The color of pigments used 
in paints and coloring materials is due to their power of 
absorbing incident light, similar to that shown in § 330 in 
the case of colored cloth and paper. The mixing of pig- 
ments is a very different thing from mixing lights or 
colors. ^ A striking example is furnished by pigments that 
reflect the complementary colors blue and yellow. 

Pulverize pieces of yellow and blue crayon, but keep the 
powders separate. If now about equal portions of the two 
powders be thoroughly mixed together, a bright green 
appears. 

The cause of the phenomenon is the imperfect absorption 
of the pigments. The yellow powder subtracts from white 
light all except yellow and green^ and the blue powder sub- 
tracts all except blue and green. Hence the only color not 
absorbed by one powder or the other is green. 

2. SPECTRA 

334. The Solar Spectrum. — The spectrum of sunlight, 
or the solar spectrum, frequently presents itself in nature 
in the rainbow. In the production of the rainbow the sun- 



LIGHT: COLOR AND SPECTRA 



325 



light is dispersed by spherical raindrops. Light from the 
sun strikes a drop at Ay Fig. 262, and is refracted to B. 
At jB a portion of the 
light is reflected to (7, 
where it emerges and 
enters the eye U. At 
the points A and O dis- 
persion accompanies re- 
fraction, and the red 
light takes the direction 
rrr, while the violet fol- 
lows the path vvv, A 
study of Fig. 268 will 
show that the eye re- 
ceives the red from a greater angle of elevation than the 
violet ; hence red lies at the outside of the rainbow. 




Fig. 262. — Dispersion of Sunlight by 
a Raindrop. 




Fig. 263. — Showing the Spectrum Colors in their Relative Positions in the 

Rainbow. 

The secondary rainbow often accompanies the primary 
one. In this case, as Fig. 264 shows, light suffers two 



326 



A HIGH SCHOOL COURSE IN PHYSICS 



reflections within the drop and emerges with the violet at 
the greater angle of elevation ; hence here in the secondary 
bow the red band is the inner circle of color. 




Fig. 264. — Showing the Formation of a Secondary Bow. 

335. Absorption of Light by the Medium. — Let a narrow, 

horizontal beam of sunlight come into a darkened room through a 
vertical slit S, Fig. 265, and pass through a convex lens L to the 
carbon disulphide prism P. (The dispersion produced by a glass 



Sunliaht 



I 




Fig. 265. — Method of Projecting the Solar Spectrum 

prism is too small to give satisfactory results.) Reflect some of the 
light from the back of the prism to the screen A B where the spec- 
trum is to be formed, and adjust the lens until it focuses a sharp 



LIGHT: COLOR AND SPECTRA 327 

image of the slit on the screen. A good spectrum can now be produced 
by setting the prism in the position shown in the figure. 

While the spectrum is on the screen, place before the slit at C a 
flat tank or bottle containing a dilute solution of chlorophyl, made by 
soaking a quantity of grass in warm alcohol. It will be observed 
that a broad dark band is formed in the red portion of the spectrum 
and a less marked band appears in the green. 

It appears from this experiment that a certain portion 
of the solar spectrum may be removed by the absorption 
of certain wave lengths in the medium through which the 
light passes. The resulting spectrum is called an absorp- 
tion spectrum to distinguish it from the continuous spec- 
trum that is obtained when none of the light is absorbed. 
Continuous spectra are, in general, given off by all lu- 
minous solids and liquids. It will appear presently that 
the solar spectrum is an absorption spectrum. 

336. The Solar Spectrum in Detail. — The most interest- 
ing and remarkable features of the solar spectrum are not 
revealed in the rainbow, nor in the spectrum ordinarily 
projected by a prism, on account of the overlapping color 
bands. In order to produce a pure spectrum in which the 
colors are more distinctly separated than before, it is nec- 
essary to work with a narrow slit which is very sharply 
focused by a lens on a white screen. 

Let the solar spectrum be projected as before, but make the slit 
about one half a millimeter in width and use a long focus (about 
50 centimeters) lens. Place the screen about 1 meter from the lens and 
prism. By making all the adjustments with care, it will easily be 
seen that the spectrum is crossed vertically by many dark lines. By 
moving the screen back and forth the best position for showing the 
lines can readily be found. 

These dark lines across the solar spectrum are known as 
Fraunhofer lines in honor of the German astronomer 
who first mapped them out about 1814. When the spec- 
trum is magnified sufficiently, hundreds of these lines 



328 A HIGH SCHOOL COURSE IN PHYSICS 

become visible. The presence of the Fraunhofer lines 
shows that certain wave lengths are either absent 
entirely from sunlight or that they are so weak as to 
appear dark by contrast. The solar spectrum is thus 
observed to be an absorption spectrum. 

337. Fraunhofer Lines Produced by Absorption. — The 
explanation of the dark lines of the solar spectrum is due 
to Stokes and Kirchhoff, the former an English, the latter 
a German, physicist. The theory is based on the follow- 
ing experiment: 

Project a well-defined solar spectrum as in the preceding experi- 
ment. Now sprinkle common salt (sodium chloride) on the wick 
of an alcohol lamp, place the lamp Just below the slit at C, Fig. 265, 
and ignite the alcohol. A bright yellow light will be emitted by 
the volatilized salt, but on the screen will appear a darkening of a 
Fraunhofer line in the yellow part of the spectrum. 

It thus appears that when light passes through the so- 
called sodium flame, the absorption of a certain wave 
length takes place. In the same manner the light from 
the white-hot central mass of the sun, which alone would 
give a spectrum without dark lines, passes through sur- 
rounding vapors, each of which, like the sodium vapor, 
has the power of removing light of certain wave lengths. 
The wave lengths absorbed by a heated gas or vapor are pre- 
cisely those which the 
vapor itself is capable of 
emitting. 

338. Bright-line Spec- 
tra. — Prepare an alcohol 
lamp for producing yellow 
light as in the preceding 
Fig. 266. — Apparatus Arranged for Bright- experiment. Place the lens 
line Spectra. j^^ ^ig. 266, so that the slit 

A is at its principal focus, and set the prism as shown in the figure. 
Focus a small telescope T on a distant object, and then place it for 




LIGHT: COLOR AND SPECTRA 329 

receiving light from the prism. Place the lamp at F, and ignite 
the alcohol. If all is properly adjusted, a bright double yellow line 
will be visible in the telescope. This is the spectrum of incandes- 
cent sodium vapor. Repeat the experiment by using strontium 
chloride in another lamp. Red and blue lines should appear. By 
using calcium chloride, red and green lines become visible. The 
yellow sodium spectrum will always appear, since sodium exists as an 
impurity in the lamp wick and in practically all salts. 

These experiments show that incandescent vapors emit 
light in which only certain wave lengths are present. The 
spectra of such bodies are therefore bright-line spectra. 
Since these spectra are characteristic of the chemical ele- 
ments, many substances can be identified by the wave 
lengths of the light which their incandescent vapors emit. 
The spectra of the chemical elements are as well known as 
their densities, specific heats, and other physical and chemi- 
cal properties. Several new elements have been discovered 
by the observation of bright spectral lines which could not 
have been produced by any known substances. An incan- 
descent vapor also affords a convenient means of obtaining 
light of one wave length or color, i.e. monochromatic light. 

339. Solar Elements. — Since the spectra of many incan- 
descent vapors have been examined and compared with 
the Fraunhofer lines of the solar spectrum, most of these 
lines have been accounted for just as the existence of 
incandescent sodium vapor surrounding the sun accounts 
for the dark sodium line. (See § 337.) If the corre- 
spondence between lines of the solar spectrum, shown in 
the middle in Fig. 267, and the spectrum of iron vapor, be 
noted, there can be no doubt as to the existence of iron 




Fig. 267. — Showing the Coincidence of the Bright Lines of the Spectrum of 
^ Iron with Some of the Dark Lines of the Solar Spectrum. 



330 A HIGH SCHOOL COURSE IN PHYSICS 

in the sun. Other solar elements are calcium, hydrogen, 
sodium, nickel, magnesium, cobalt, silicon, aluminium, 
titanium, chromium, manganese, carbon, barium, silver, 
zinc, and many others. It is an interesting fact that the 
element helium was first discovered in the sun by Sir 
Norman Lockyer, of England, by means of its dark lines 
in the solar spectrum. Later, in 1895, Sir William 
Ramsay discovered an element that gave the same spec- 
trum, and which was therefore given the same name. 
Helium is at present a comparatively well-known gas that 
can be produced in all chemical laboratories. 

The instrument by which the spectra of celestial and 
terrestrial objects are produced is called the spectroscope. 
By combining the spectroscope with the telescope, astrono- 
mers have succeeded in ascertaining to some extent the 
composition of man}^ stars and in collecting other informa- 
tion of great scientific value. 

EXERCISES 

1. What kind of a spectrum should moonh'ght give? 

2. What kind of a spectrum would you expect to obtain by dis- 
persing the light from a live coal by means of a prism ? 

3. The luminosity of an oil or gas flame is due to heated particles 
of carbon. What should be the nature of the spectrum of a kerosene 
lamp flame ? 

4. What is the position of the sun relative to falling rain and an 
observer when a rainbow is seen? 

5. Why do colored fabrics often appear different when viewed by 
artificial light ? 

6. If the waves producing the sensation of red were all absorbed 
from sunlight, what color would remain? What color would objects 
that were formerly red appear to have ? 

3. INTERFERENCE OF LIGHT 

340. Color Bands by Interference. — We have seen how 
composite light can be separated into its component colors 



LIGHT: COLOR AND SPECTRA 



331 




by means of a prism, but there 
still remain a great many cases 
of color formation which are not 
at all due to dispersion or ab- 
sorption. 

Let two strips of plate glass, A and 
B, Fig. 268, about 1 inch wide and 5 
inches long, be separated at one end by 
a piece of tissue paper C (exaggerated 
in cut) and the other ends clamped 
tightly together. (The plate glass 
covers accompanying sets of small 
weights may be used.) Produce a yel- ^j^. 268. - Alternate Dark and 
low sodium flame by bringing in con- Bright Bands Produced by In- 

tact with a Bunsen or alcohol flame a terference of Light Waves, 

piece of asbestos D, wet with a solution of common salt. Now hold 
the glass strips behind the flame, and observe the images produced. 
A series of dark and yellow bands will be seen extending transversely 
across the glass. 

The explanation of 
this interesting phe- 
nomenon depends on 
the wave theory of 
light. The flame, as 
we know, sends out 
only yellow light 
(§338). This is in 
part transmitted and 
in part reflected at 
each glass surface 
(§ 282). Let ^^ and 
AC, Fig. 269, repre- 
sent the interior sur- 
faces of the plates much 
enlarged, and let the 

Fig. 269. — Diagram Showing Points of In- 
terference and Reenforcement. wave line ab represent 




Interference. 



d Re-enforcement. 



Interference, 



fie-enforcement 



332 A HIGH SCHOOL COURSE IN PHYSICS 

the light reflected at the point a on the surface AC. 
Also let the dotted line a'h represent the waves reflected 
at a' on the surface AB. These two trains of waves ob- 
viously interfere and destroy each other just as do two 
trains of sound waves when they coincide in this manner 
(§ 190). Therefore, since at this place on the glass 
plates the light waves cancel each other, we see a dark 
band. But at the point e the case is different. Since 
the thickness of the air film is here a quarter of a wave 
length greater than at a, the train of waves reflected 
from c' coincides with that reflected at e in such a manner 
as to produce reenforcement. Hence at this point we see 
the bright yellow band. Similarly the wave trains re- 
flected at e and e' interfere and cancel, while at g and g' 
we again find reenforcement. Thus the dark and bright 
bands appear alternately in the plates. 

341. Color Bands in Soap Films. — The following ex- 
periment can easily be performed by letting a soap film take 
the place of the wedge-shaped air film of the preceding 
experiment. 

Let a film of soapy water be produced on a wire loop as described 
in § 130. Hold the film behind a yellow sodium flame, and observe 
the image in the film by reflected light. Many narrow bands of 
yellow will appear which continue to grow broader and farther apart 
until the film breaks. 

When the wire loop is held in a vertical position, the 
liquid runs slowly down from the top, forming a wedge- 
shaped film that is thinnest at the upper edge. One por- 
tion of the yellow incident light is reflected from the 
front surface, and another portion from the back, after 
traversing the thickness of the film. Hence the portion 
reflected from the back passes through the film twice. 
Now if the two reflected trains interfere, a dark band is 



LIGHT: COLOR AND SPECTRA 333 

produced ; but if they reenforce, a bright band of yellow 
appears, precisely as described in the preceding section. 

342. White Light Decomposed by Interference. — The 
result of the experiment described in § 341 leads to the 
explanation of the beautiful coloration produced when 
the sunlight falls on soap bubbles, oil films on water, etc. 

Let sunlight fall upon the plates of glass arranged as in § 340. 
Variegated bands occurring alternately will appear instead of the 
yellow bands obtained when sodium light is used. 

On account of the fact that sunlight, or white light, is 
a composite of different wave lengths, the interference 
of the reflected waves of red, for example, takes place at 
a different point from that of the yellow. When the 
waves of red are canceled by interference, the comple- 
mentary color, or bluish green, is left. At the point where 
waves of yellow interfere, its complement, or blue, appears. 
Thus bands consisting of the complements of all the 
spectral colors are produced. 

SUMMARY 

1. White light is decomposed into the spectral colors by 
passing through a triangular prism of glass, water, ice, etc. 
The cause of dispersion lies in the fact that different wave 
lengths are retarded different amounts, and hence are 
refracted in differing degree by the prism. Long waves 
(red) suffer the greatest refraction, while the shorter 
waves (violet) suffer least (§§ 827 to 329). 

2. The color of an object depends both on the light 
which falls upon it and that which it reflects to the eye. 
Various wave lengths are absorbed by all except white 
bodies (§ 330). 

3. The color of a transparent body depends on the color 
of the light which it transmits (§ 331). 



334 A HIGH SCHOOL COURSE IN PHYSICS 

4. Complementary colors are those which when added 
in proper proportions produce white (§ 332). 

5. The color of pigments used in the manufacture of 
paints, etc., is due to the absorbing power of the material 
(§ 333). 

6. The solar spectrum is the spectrum of sunlight. The 
rainbow is an example. The pure spectrum is crossed by 
numerous dark lines produced by the absorption of certain 
wave lengths by heated gases in the solar atmosphere. 
These lines reveal the chemical elements in the sun's com- 
position (§§334 to 339). 

7. Interference of light is brought about by the coinci- 
dence of waves in such a manner as to weaken or cancel 
each other. The alternate interference and reenforcement 
of light waves in thin wedge-shaped films gives rise to the 
colors seen in oil films, soap bubbles, etc. (§§ 340 to 342). 



CHAPTER XVI 



ELECTROSTATICS 



1. ELECTRIFICATION AND ELECTRICAL CHARGES 

343. An Electric Charge Produced. — It is a matter of 
common observation that a hard rubber comb acquires the 
power of attracting bits of tissue paper and other very 
light bodies simply by being drawn through dry hair. The 
comb may also be put into this condition by being rubbed 
with silk or flannel. Knowledge of this phenomenon 
dates from the Greeks of 600 B.C., when it was known 
that amber would attract light objects after having been 
rubbed with silk. No advance was made in the science 
of electricity until the time of Sir William Gilbert of 
England, about 1600 a.d. At this time Gilbert found 
that a great many substances, as glass, ebonite, sealing 
wax, etc., could be given this power of attraction by 
rubbing with fur, wool, or silk. 

Test rods of glass, ebonite, sealing wax, etc., for the power of 
attracting small pieces of dry pith before and after being rubbed 
with silk, fur, and flannel. 
See Fig. 270. 

When a body pos- 
sesses the property of 
attracting light objects^ 
as hair^ pith balls^ and 
bits of paper, it is said 

to be electrified The ^^^* ^^^" — Action of an Electrified Glass Rod. 

change is brought about by a charge of electricity. The 
process by which a body is electrified is called electrifica- 

335 




336 



A HIGH SCHOOL COURSE IN PHYSICS 




t'lG. 271. — Repulsion between Two Simi- 
larly Charged Bodies. 



tion. These terms are all derived from the Greek word 
electron^ meaning amber. 

344. Electrical Charges of Two Kinds. — Procure two large 

sticks of sealing wax and 
a glass rod. Electrify a 
stick of sealing wax by rub- 
bing it with a flannel or 
fur and suspend it in a wire 
stirrup as shown in Fig. 
271. Rub the other stick 
of sealing wax and bring it 
near the suspended one. A 

decided repulsion will be 
observed. Now rub the glass rod with silk and bring it near the sus- 
pended wax. Attraction takes place. 

The experiment shows that the electrified glass and 
sealing wax cannot be in precisely the same condition, 
since they act differently toward the suspended body. 
This difference in the behavior of electrified bodies is 
said to be due to the kind of charge developed when they 
are rubbed. Thus glass is said to be charged positively 
when rubbed with silk ; and sealing wax, negatively when 
rubbed with flannel or fur. Some specimens of glass are 
electrified negatively when rubbed with cat's fur or flannel. 

345. A Law of Electric Action. — The preceding experi- 
ment shows that two sticks of sealing wax repel each 
other after being charged negatively. In the same man- 
ner two positively charged glass rods will show a repulsion. 
But any negative charge will attract a positive charge, 
just as the glass attracts the suspended sealing wax when 
both are electrified. Hence, we may infer that electrical 
charges of a similar kind repel each other ^ and those that are 
dissimilar attract. 

346. The Electroscope. — In order to detect the presence 
of a charge upon a body, an instrument called the electro- 



ELECTROSTATICS 



337 




Fig. 272. — The 
Gold-leaf Elec- 
troscope. 



scope is employed. The gold-leaf electroscope is shown in 
Fig. 272. This instrument consists of a metal rod which 
penetrates the rubber stopper of a flask. At the top the 
rod terminates in a plate or ball, and at 
the lower end it is provided with two 
strips of gold foil. Under ordinary con- 
ditions the leaves hang parallel ; but when 
an electrical charge is brought near, they 
diverge and thus show the presence of 
electrification. 

347. Determining the Kind of Charge. 
— The electroscope is frequently used to 
ascertain the kind of charge on an elec- 
trified body. The following experiment will show how 
this can be done. 

Make a proof plane by sealing a small coin or tin disk to the end 
of a small glass rod to be used as a handle. Touch the disk of the 
proof plane to a positively charged glass rod, and then to the plate 
of the electroscope. The divergence of the leaves will show that the 
instrument is charged. Now, if the charged glass rod is carefully 
brought near the electroscope, the divergence will increase; but if 
the negatively charged sealing wax is brought near, the divergence 
at once decreases. While the electroscope is thus charged, bring up 
an electrified rod of ebonite or a charged rubber comb, and note 
whether the divergence increases or decreases. In this case the charge 
on the ebonite will be found to act like that on the sealing wax and 
hence produce a decreased divergence of the leaves. 

In a word, the nature of an unknown charge is deter- 
mined by observing whether its effect on a charged electro- 
scope is like that of a positively charged glass rod or the 
negatively charged sealing wax. 



Fig. 273. — Opposite Charges 
Developed at the Same 
Time. 

23 



348. Positive and Negative 
"^ Charges Developed Simultane- 
ously. — Let a flannel cap about 6 
inches long be made that will fit closely 



338 



A HIGH SCHOOL COURSE IN PHYSICS 



over the end of an ebonite rod as in Fig. 273. A represents a silk 
thread by which the cap may be handled. Now turn the rod in the 
cap to electrify it and, without removing the cap, hold the rod near 
the plate of the electroscope. Little or no charge will be detected. 
Remove the cap by means of the thread A and present it to the elec- 
troscope. When tested as in § 347, the cap will be found to be charged 
positively. If the rod be now tested, a negative charge will be found. 

On account of the fact that the rod and cap together 
produce no effect on the electroscope, we may infer that 
their charges exactly cancel. The conclusion is, there- 
fore, that when a certain quantity of one kind of electrifica- 
tion is produced hy rubbing a rod^ an equal amount of the 
opposite hind appears on the object with which it is rubbed. 

349. Charging by Contact. — By 

means of a silk thread or silk fiber suspend 
a small ball of dry elder pith as shown in 
Fig. 274. Hold near the ball a positively 
charged glass rod. The ball is at first at- 
tracted to the lod and then violently repelled. 
The repulsion shows that the ball by contact 
with the rod has been charged with a kind of 
electrif cation similar to that in the rod. In 
this case the charge is positive. Likewise the 
ball will be charged negatively by contact 
with a negatively charged body. 




Fig, 274. — Experiment 
with a Suspended 
Pith Ball. 



The experiment shows clearly that when an insulated 
body comes in contact with a charged body, it becomes 
charged with electricity of the same kind as that on the 
charged body. In this way the proof plane used in § 347 
carried from the glass rod to the electroscope an electrical 
charge of the same kind as that on the glass. 

350. Conductors and Insulators. ^ — Select a number of pieces 
of wood, metal, glass, ebonite, cardboard, leather, etc. Let one end of 
a metal rod be supported on the electroscope and the other end on an 
ebonite rod A, Fig. 275. Bring a charged body R against the end of 
the rod opposite the electroscope. A sudden divergence of the leaves 



ELECTROSTATICS 339 

shows the transfer of a charge from R to the instrument. Experi- 
ment with a rod of wood in the same manner. The divergence, if any 
is produced, takes place more slowly than before if the wood is dry. 
Glass and ebonite should be found 
not to transfer any appreciable _======^^^7'/''^ 

charge unless their surfaces are 




moist. 

Some substances have the 

power to transfer charges of Fig 275. — Testing the Conductivity 

electricity, while others do of Rods, 

not. Metals are shown to be good conductors ; dry wood 
is a poor conductor, while glass and ebonite are practically 
non-conductors. Non-conductors are called insulators. 
Among the best insulators may be mentioned dry air, 
glass, mica, shellac, silk, rubber, porcelain, paraffin, and oils. 
Substances of all degrees of conductivity exist, varying 
from the best conductors down to the best insulators. 

EXERCISES 

1. Place a piece of paper against the wall and stroke it with cat's 
fur. It will be found to adhere to the wall for several minutes. 
Explain. 

2. Why is it more difficult to brush lint from clothing in cold dry 
weather than at other times? 

3. Draw a rubber comb through dry hair and test the charge de- 
veloped on the comb. Now present a positively charged rod to the 
charged hair and observe whether it is repelled or attracted. Com- 
pare this experiment with that of § 348. 

4. After walking on a silk or woolen rug or over a glass floor, one 
often finds that sparks will jump between the finger and a gas fixture 
near which it is held. The gas may even be lighted in this manner. 
Explain. 

5. Support a dry pane of glass about 1 in. above a quantity of 
small pieces of dry pith. Rub the glass with silk and explain the 
agitation of the pith observed. 

6. Balance a meter stick on the smooth bottom of a flask and see 
if it is alfected by a charged glass rod. Are only very light bodies 
attracted ? 



340 



A HIGH SCHOOL COURSE IN PHYSICS 



2. ELECTRIC FIELDS AND ELECTROSTATIC INDUCTION 

351. Lines of Force. — We have already observed that 
an electrified body has an effect upon another body even 
when placed at a distance of several inches. Light objects 
will rise from the table toward a charged glass rod, and 
the leaves of an electroscope will often show a divergence 

at a distance of four or five 
feet from the charge. The 
space in which an electric 
charge affects surrounding 
objects is called the electric 
field due to that charge. 
If lines are drawn in an 
electric field to represent 
at every point the direction 
in which a charge, as A, 
Fig. 276, tends to move 
a very small, positively 




Fig. 276. —Lines of Force Emanating 
from a Cliarge. 



charged body B, they are called lines of force. Thus every 
electric field is assumed to be filled with lines of force. 

Lines of force extend from positive charges to nega- 
tive charges, as shown in Fig. 277. In other words, a 
positive charge always exists at 



\ 






— -^v\\| 



one end of a line of force and , ^ ^^ ^ 

a negative one at the other. J^^^tJiJ'^^' ~^'^fy-<'^^ 

Therefore, since each line of ^/ /'<^§^::~~--''-^Wi\'\\'"C^ 

. / ' ' ^ \^> — '>y I I \ ^ 



force must have two extremi- 



\ 



ties, for every positive charge Fig. 277. — The Electric Field be- 
,1 ,1 1 tween Two Unlike Charges. 

there must be somewhere a cor- 
responding negative charge. This fact gives rise to the 
phenomena shown in the following sections. 

352. Electrification by Induction. — Let a charged glass rod 
be brought near a gold-leaf electroscope. The leaves begin to diverge 



ELECTROSTATICS 



341 



when the rod is at a distance and spread piore and more as the rod 
approaches. On removing the rod the leaves fall together again. 

The temporary effect produced in the electroscope, due 
to the presence of a neighboring charge, is the result of 
electrostatic induction. The leaves of the instrument are 
obviously affected only while they are in the electrical 
field around the charged rod, since they collapse as soon 
as the rod is removed. In this case there is no change in 
the amount of electrification on the glass rod, and the 
effect can easily be shown to take place through an inter- 
vening plate of glass or other insulating material. Hence, 
in the process of electrostatic induction a transfer of elec- 
tricity from one body to the other does not take place. 

353. Electrical Separation by Induction. — The condition 
of an electroscope which is placed near a charged body, as 
in the preceding section, is readily understood after the 
following experiment : 

Place two metal vessels A and B, Fig. 278, in contact on an elevated 
block of paraffin, or some other good insulating material. Each ves- 
sel should be provided with 
a small pith ball attached 
to the top by means of a 
piece of cotton thread. Now 
bring a positively charged 
glass rod R near one of the 
vessels, as B, and, by means 
of the silk thread C, sepa- 
rate the vessels before re- 
moving the rod. When the 
rod is taken away, the pith 
balls will show that each 




Fig. 278. — Separation of Positive and 
Negative Charges. 



vessel has acquired a charge, but the charge on the rod R has not been 
diminished. With the help of the proof plane and electroscope test 
each charge ; that of B will be negative, that of A positive. 

From this experiment it is clear that whenever an electri- 
fied body is brought near an unelectrified insulated conductor^ 



342 



A HIGH SCHOOL COURSE IN PHYSICS 



the opposite hind of electrification is induced on the nearer 
side and the same kind on the remote side. 

354. Charging by Induction. — An insulated metallic 
body may be charged by making use of the separation of 
positive and negative electricity, as shown in the preceding 
section. 

Hold the positively charged glass rod about 6 in. from the plate of 

the gold-leaf electroscope. The leaves 
diverge because the positive charge 
on the rod induces a negative charge 
in the plate and a positive charge in 
the leaves, as shown in Fig. 279. 
While the rod is in this position, 
touch the electroscope with the finger. 
The leaves collapse. Now remove the 
finger and then the rod. The leaves 
diverge again, thus showing that a 
charge is left on the instrument. If 
this charge be tested by bringing a negative charge near the elec- 
troscope, an increased divergence will show that it is negative. 

355. Explanation of the Process. — The electrical con- 
ditions that exist during the process of charging a conduc- 
tor by induction are represented in Fig. 280. The presence 




+ / ^ + 

FiG. 279. — Charged Rod Acting 
by Induction on an Electro- 
scope. 





(2) 



(3) 



Fig. 280.- Illustrating the Process of Charging by Induction. 



of the positively charged rod H produces a separation of 
positive and negative electricity in the conductor A (§ 353), 
just as if an uncharged body possessed equal amounts of 
these two kinds, as shown in (1). Lines of force extend 
from the positive on the rod to the induced negative on 



ELECTROSTATICS 



343 



the conductor, while other lines extend away from the 
positive at the remote end to the walls of the room, 
near-by objects, etc. When the conductor is touched with 
the finger, as in (2), the repelled or "free" charge, which 
in this case is positive, is permitted to escape through the 
body to the earth. The negative or " bound " charge re- 
mains on the conductor on account of the influence of the 
charged rod R. On removing first the finger and then 
the rod, the negative charge 
distributes itself over the con- 
ductor, as shown in (3). If R 
is a negatively charged body, 
the signs of all the charges will 
simply be reversed. 

356. Distribution of Electric- 
ity on a Conductor. — Insulate a 
metal vessel A, Fig. 281, by placing 
it on a dry tumbler 5, or a plate of 
paraffin or beeswax. Charge the ves- 
sel as highly as possible and then 
try to take charges from its interior 
surface to the electroscope by means 
of a proof plane c. It will be found 
that no charge can be obtained in 
this vi^ay. Now try to take a charge 
from the exterior surface of the vessel. 
that this attempt is successful. 




Fig. 281. — No Charge can be 
Taken from the Interior Sur- 
face of a Charged Body. 

The electroscope will show 



This experiment shows very conclusively that an elec- 
trical charge distributes itself over the exterior surface of 
an insulated conductor. This is just the result that is to 
be expected if we consider that the various parts of any 
charge are mutually repellent. On this account they will 
separate to the greatest extent possible, which is the case 
when the charge is distributed over the exterior surface of 
the body charged. 




Fig. 282. — Elec- 
trical Den- 



344 A HIGH SCHOOL COURSE IN PHYSICS 

357. Distribution of a Charge not Uniform. — Charge a large 
insulated egg-shaped conductor, Fig. 282. By means of a proof plane 
transfer a charge from the large end of the body to 
the electroscope and observe the divergence of the 
leaves. Now discharge the electroscope and test the 
small end of the conductor in a similar manner. A 
greater divergence of the leaves will be obtained. 



It thus appears that an electrical charge 

distributes itself over a conducting surface 

sity is Great- in accordance with the shape of the body. 

est at the rr^-i ,-i j_- ^ • 

Pointed End ^^^^ quantity per square centnneter is 
of this Con- greatest at the small end. In other words, 
the electrical density is greatest where the 
conductor is the most sharply pointed. In fact, if a sharp 
point be attached to the charged conductor, the density at 
the point may be so great that the charge will be sponta- 
neously discharged into the surrounding air. 

358. Effect of Points. — By the help of sealing wax or shellac 
attach the center of a sharp needle to a glass handle. Bring a charged 
glass rod over the electroscoj^e, and, while the rod remains in this 
position, bring the eye end of the needle against the plate of the 
electroscope and the point toward the charged rod. Now remove 
both needle and rod, and the electroscope will be positively charged. 
Again, hold a charged glass rod near an insulated conductor, as a 
metal vessel, and then place the eye end of the needle against the 
opposite side of the conductor. Remove the needle and then the rod 
and test the conductor for a charge. A negative charge will be found. 

The results of these experiments depend upon the electri- 
cal discharge from pointed conductors. On account of 
the great electrical density at a point, an intense field of 
force exists in the immediate neighborhood. Air particles 
are forcibly drawn against the point and charged by con- 
tact with the same kind of electricity, after which they are 
violently repelled. Thus a so-called electrical wind is set 
up which conveys away the charge at a rapid rate, 



ELECTROSTATICS 345 

359. Lightning Rods. — Positive evidence regarding the 
identity of lightning and electrical discharges was secured 
by the classical experiments of Benjamin Franklin^ in 1752, 
when he succeeded in drawing an electrical charge from a 
thunder cloud along the string of a kite. Through his 
suggestion lightning rods were first used as a protection for 
buildings against damage by lightning. The principle has 
been demonstrated in the preceding section. A strongly 
charged cloud passes over a building, and between the cloud 
and the building there is set up an intense field of force 
(§ 351). If this field becomes of sufficient intensity, the 
air, being no longer able to insulate, breaks down as an 
insulator. The sudden discharge tears the roof of the 
building, and the intense heat often produces a conflagra- 
tion. The presence of a pointed conductor, however, lead- 
ing from above the roof to the earth, permits a gentle and 
harmless discharge of electricity to take place between the 
cloud and the earth. The effectiveness of lightning rods 
is often impaired by the use of dull points and poor ground 
connections. 

The thunder that accompanies a lightning flash is caused 
by the impact of the air as it is forced in to fill the partial 
vacuum which is developed along the line of electrical dis- 
charge. At a distance from the discharge, the direct 
report is followed by echoes from the clouds and woods, 
causing the rumblings so common in most localities. 

EXERCISES 

1. In testing a charge, why is it necessary to work with a charged 
electroscope ? 

2. When a gold-leaf electroscope is charged with negative elec- 
tricity, for example, why will the approach of a positively charged 
body produce first a decrease and then an increase of the divergence 
of the leaves ? 

1 See portrait facing page 346. 



346 



A HIGH SCHOOL COURSE IN PHYSICS 



3. If several insulated metal vessels are placed in a row, but not in 
contact with each other, what will be the electrical condition of each 
when a positively charged rod is held near one end of the row? Illus- 
trate by means of a diagram showing four such vessels. 

4. If the vessel at the end of the row near the charged rod is 
touched with the finger and then both finger and rod are removed, 
what will be the electrical condition of each vessel ? Illustrate this 
case by a diagram. 

5. Represent by diagrams the fields of force existing under the 
conditions described in Exer. 3. 

6. Represent the fields of force as they exist under the conditions 
given in Exer. 4 and compare each field with the corresponding one 
of the preceding exercise. 



A 



3. POTENTIAL DIFFERENCE AND CAPACITY 

360. Electrical Flow. — It has already been observed in 
§ 349 that a charge of electricity can be transferred by a 
conductor from an electrified body to the electroscope. 
The effects that accompany such a transmission of elec- 
tricity, as will be seen later, enable it to be employed as 
one of the most important agents under the control of man. 
The following experiment will bring out more clearly the 

conditions under which a 
transfer of electricity takes 
place. 

Provide two similar metal ves- 
sels A and B, Fig. 283, with pith 
balls and insulate them on blocks 
of paraffin. Let each vessel be 
positively charged, but A more 
highly than B, as measured by the 
repulsion of the pith balls. Now 
connect the vessels by means of a wire attached to a sealing wax 
handle C. The pith ball on A will fall slightly while that on B will 
rise. Thus some of the charge on A moves along the wire to B. 

Although both insulated vessels were originally charged 
with the same kind of electricity, in the language of 



+ 



+ 



m 



Fig. 283. — Conductor A has a Higher 
Potential than B. 




BENJAMIN FRANKLIN (1706-1790) 

The achievements of Franklin in the field of electricity are no 
less brilliant than his successes as a statesman and diplomat. His 
attention was first turned to the study of electrical phenomena by 
witnessing some experiments which at that time (1746) were re- 
garded as no less than marvelous. His experiment to prove the 
electrical nature of lightning has become classic. In order to ascer- 
tain whether electricity could be obtained from clouds during a storm, 
Franklin constructed a kite which was provided with a pointed metal 
rod for " drawing off " the electrical charge, if there should prove 
to be any. At the approach of a storm, Franklin and his son raised 
the kite, which was held by a hempen cord. The lower end of the 
cord was tied to a metal key through which was passed a ribbon 
of silk to protect the body from severe and dangerous shocks. As 
soon as the cord became moistened by the rain, electric sparks were 
readily drawn from the key. In regard to this experiment, Franklin 
writes: "At the key the Leyden jar may be charged; and, from the 
electric fire thus obtained, spirits may be kindled and all other 
electrical experiments performed which are usually done by the 
help of a rubbed glass globe or tube, and thereby the sameness of 
the electrical matter with that of lightning completely demonstrated." 

" Antiquity would have erected altars to this great and powerful 
genius who, to promote the welfare of mankind, comprehending both 
the heavens and the earth in the range of his thought, could at once 
snatch the bolt from the cloud and the scepter from tyrants." — 

MlRABEAlT 



ELECTROSTATICS 



347 



VaWWa 



A 



+ 


A 


+ 




- 


[\ 


a' 




b' 




c' 




+ 




+ 




- 





-K 



Fig. 



Physics, A was charged to a higher potential than B. Thus 
a charge moves from a point of higher to one of lower poten- 
tial^ just as lieat flows along a heat conductor from a point 
of higher to one of lower temperature. 

361. Potential Difference. — It has just been observed 
that an electrical charge will flow along a conductor as 
long as there exists a potential difference. It is precisely 
this difference of poten- 
tial that determines 
whether electricity will 
flow and the direction 
it will take. Figure 
284 will show to some 
extent the differences 
that may exist between 
electrical charges. AB 
represents the level 
ground, and a, 6, <?, and 
d four tanks containing water. Tank a is analogous to 
an insulated body charged positively to a high potential, 
as vessel a\ while tank h represents one of a lower poten- 
tial than a\ as h' . It is assumed that all positive charges 
are of a higher potential than the earth and will discharge 
to the earth unless insulated from it. On the other hand, 
negative charges, as c^ and d' ^ are assumed to have lower 
potentials than the earth. The potential of the earth is 
thus on the dividing line between positive and negative 
charges; hence, the potential of the earth is regarded as 
zero. 

Connecting any two of the tanks shown in the figure 
would evidently result in a transfer of water from left to 
right ; thus joining any two of the charged vessels by 
means of a conductor would bring about a transfer of 
electricity in the same direction. 



284. — Analogy between Electrical 
Potential and Water Level. 



348 A HIGH SCHOOL COURSE IN PHYSICS 

362. Mixing Positive and Negative Charges. — Let the two 

insulated metal vessels used in § 360 be charged with equal amounts 
of positive and negative electricity as shown by the divergence of the 
pith balls. Now connect them by means of a wire provided with an 
insulating handle as before. Both balls fall against the sides of the 
vessels, showing that the two charges have neutralized each other. 
Repeat the operations just described after charging A positively until 
its pith ball diverges considerably more than that of B, which is 
charged negatively. After the connection has been made between 
the vessels, a positive charge will be found on each vessel. 

Not only does electricity tend to flow from a positively 
to a negatively charged body, but a mixture of unlike 
charges tends to produce a cancellation of both. If, how- 
ever, one of the charges exceeds the other in amount, the 
excess remains distributed over both surfaces. In the 
final condition both bodies are of the same potential. 

363. The Electrostatic Unit of Quantity. — Unit charges 
of electricity are such equal quantities as exert upon each 
other a force of one dyne (§ 35) when separated hy one 
centimeter of air. The force, as we have seen, is repellent 
when the charges are of the same kind and attractive 
when they are different. If two equal quantities of posi- 
tive and negative electricity are mixed, they cancel each 
other; but if both of the two mixed charges are of the 
same kind, the resulting charge is tlieir sum. 

Example. — If 10 units of positive electricity are mixed with 12 
units of negative, what will be the result? 

Solution^ — The 10 positive units will cancel 10 negative units, 
leaving 2 units of negative electricity, which will be distributed 
over the surface of the conductors. 

364. Electrostatic Capacity. — By means of a proof plane 
transfer a charge from an electrified glass rod to the electroscope and 
note the approximate divergence of the leaves. Now place one of the 
metal vessels used in § 353 upon the plate of the electroscope and again 
transfer a charge with the proof plane as before. In this case the 
divergence will be found to be much smaller than before. Continue 
to transfer charges until the divergence is the same as at first. 



ELECTROSTATICS 



349 



By placing the metal vessel upon the electroscope, the 
surface over which a charge will distribute itself is 
materially increased. Hence a given quantity of elec- 
tricity will produce a smaller electric density, and the 
mutual repulsion of the gold leaves will be lessened. On 
this account, a larger quantity will he required to produce 
the same divergence of leaves ; or^ in other words^ to produce 
the same potential as at first. The change produced in the 
condition of the conductor is expressed by saying that the 
electrostatic capacity is increased by the increased area 
presented by the metal vessel. 

365. Condensers. — In many of the practical applica- 
tions of electricity, it is necessary to make use of some 
device that has many times the capacity of anything we 
have used in the preceding experiments. The manner in 
which the desired result is accomplished is made clear by 
the following experiments : 

1. Place a flat metal plate C, Fig. 285, having well-rounded cor- 
ners, upon the plate of the gold-leaf electroscope and charge the in- 
strument with the glass rod and proof plane. Now bring a similar 
metal plate B, which is provided with a handle 
of sealing wax A, near C, but not touching 
it. The divergence of the gold leaves will 
decrease, showing that the potential of the 
leaves has been lowered. Withdraw B and 
the potential of the leaves will rise to its 
original value. Repeat these operations while 
the fingers are allowed to remain in contact 
with B, thus connecting it with the earth. 
The divergence of the leaves becomes very 
small when B and C are near together, but 
returns to its former value when the plates 
are again separated. 

2. Mount a plate 5 in a clamp so that it 




is about ^ cm. from C and insulated. Count 



Fig. 285. — Showing the 
Effect of a Neigh- 
boring Conductor on 
Capacity. 



the number of charges that must be carried 

to C by a proof plane to produce a given divergence, i.e. to produce a 



350 A HIGH SCHOOL COURSE IN PHYSICS 

given potential. Now discharge C and connect B with the earth by 
means of a wire, and see how many charges must be transferred 
to C to produce approximately the same divergence as before. A 
large number will be required. 

These experiments show (1) that the potential that a 
given charge produces when placed on an insulated con- 
ductor depends upon the proximity of other conductors 
and whether they are " earthed " or not, and (2) that the 
capacity of a conductor is enormously increased by the 
presence of an earthed conductor placed very near, but 
insulated from it. A comhiyiation of plates separated hy an 
insulator constitutes an electrostatic condenser. The capac- 
ity of a condenser is proportional to the size of the plates 
and becomes greater as the distance between them is 
reduced. 

366. Influence of the Insulating Material. — Arrange the 

electroscope and plates as in Experiment 2 of the preceding section. 
Connect B with the earth and charge C until a moderate divergence 
of the leaves is produced. Now introduce between B and C a pane of 
glass and note the effect upon the divergence. The potential of C 
will fall when the glass is introduced, and will rise again when it is 
removed. Let the experiment be made by using beeswax or paraffin 
instead of glass. The effect is more marked than before. 

Since the introduction of another insulator, or dielectric, 
to replace the air between the plates of a condenser, re- 
duces the potential, it is obvious that it will require a 
greater chai:ge to bring the potential back to its original 
value. Hence the capacity of the condenser is increased. 
One of the best insulators used in condensers is mica, not 
only because it can easily be obtained in thin sheets, but 
because of its advantageous influence as a dielectric upon 
the electrical capacity of the condenser. 

367. Forms of Condensers. — One of the earliest forms 
of condensers is known as the Ley den jar^ from Ley den 



ELECTROSTATICS 



351 




Fig. 286. — The Ley- 
den Jar. 



in Holland, the place of its origin. It was first used in 
1745. This condenser consists of a glass jar, Fig. 286, 
which is coated with tin foil to about 
two thirds of its height on its interior 
and exterior surfaces. Through a cover 
of insulating material passes a metal rod 
terminating at the top in a ball and at 
the lower end in a chain which makes 
contact with the inner coating of the jar. 
Another form of condenser that is 
widely used is represented diagrammat- 
ically in Fig. 287. This condenser con- 
sists of a large number of sheets of tin 
foil of which alternate sheets are con- 
nected at A and the intervening sheets at B. In the 
best condensers of this type the insulating material sepa- 
rating the sheets of foil is mica ; but in cheaper forms, 

paraffined paper is used. The capacity 
of such condensers is large on account 
of the large area of tin foil and the 
extremely small distance between the 
conducting surfaces of the foil (§ 365). 
368. Charging and Discharging a Ley- 
den Jar. — In order to charge a Ley- 
den jar, the outer coating is connected with the earth by 
a metallic conductor, or the jar is simply held in the hand. 
A very imperfect earth connection is produced by setting 
the jar upon a table of dry wood. If now the knob of the 
jar be connected with some source of electricity, a charge 
is communicated to the inner surface. If this is positive, 
an equal amount of negative will be induced on the inner 
surface of the outer coating, and a similar quantity of 
positive repelled to the earth. In this manner a strain 
is set up in the glass which in some instances is suffi- 
cient to break the jar. 



Fig. 287. — Diagram of 
a Mica, or Paper, 
Condenser. 



352 A HIGH SCHOOL COURSE IN PHYSICS 

To discharge a Leyden jar, it is necessary to make an 
electrical connection between the two tin-foil coatings. 
If the charge is not large, this can be done safely by 
simultaneously touching the knob and the outer coating 
with the hands. The best method, however, is to bend a 
metal conductor in such a form that one end can be kept 
in contact with the outer coating while the other is brought 
near the knob. At the instant of discharge a bright spark 
will be seen to jump between the knob and the end of the 
conductor. 

EXERCISES 

1. Three Leyden jars are charged with — 4, — 7, and + 10 nnits 
respectively. How many units will remain in the jars after the knobs 
have been connected ? 

2. A Leyden jar that is placed on a plate of glass has a small 
capacity. Why ? 

3. If a Leyden jar be highly charged and then placed on a plate 
of glass or paraffin, the knob can be safely touched with the hand. 
In this condition only a small portion of the entire charge will be 
taken from the jar. Explain. 

4. Having a metal globe positively electrified, how could you 
electrify any number of other globes with negative electricity ? 

5. With a positively charged globe, how could you positively 
charge one of the other globes without reducing the charge on the 
first? 

4. ELECTRICAL GENERATORS 

369. The Electrophorus. — In order to produce larger 

electrical charges than those used in any of the preceding 

^^^^ experiments, the principle of induction 

^^^—J|i (§ 352) is advantageously employed. 

The simplest form of generator for this 

'""^"x purpose is the electrophorus (pronounced 

^^^^^^^^" e lek trof'o rus)^ an apparatus invented 

Fig. 288. — The by Volta,^ an Italian physicist, in 1777. 

Electrophorus. xhis instrument consists of a plate of 

1 See portrait facing page 352. 




COUNT ALESSANDRO VOLTA (1745^1837) 

The age of practical electricity began with the invention of the 
voltaic cell by Volta, an Italian, professor of physics at the Uni- 
versity of Pavia. Before his invention it was not known that a 
continuous current of electricity could be produced. In 1793, Volta 
announced to the Royal Society of London a discovery made by 
Galvani (1737-1798), professor of anatomy at Bologna. Galvani 
had observed that by joining together two diiferent metals and 
touching one to the muscle of a frog's leg and the other to the 
nerve, violent contractions were produced even after the animal's 
death. Most extravagant hopes were founded on this discovery, and 
many believed that a cure had been found for all diseases. In 
reality, the discovery paved the way to other and greater ones which 
have become a vital part of the history of the nineteenth century. 

The use of the two metals in Galvani's experiment suggested to 
Volta the basic principle of the modern cell consisting of two plates 
of different metals immersed in a liquid. This, as we know, becomes 
a continuous source of electricity. The invention was received with 
great enthusiasm. In a short time batteries made by joining several 
of the so-called " voltaic piles " were used in experimental work in 
many of the laboratories of Europe. With the advent of this means 
of generating electric currents began the series of discoveries that 
have led up to the phenomenal use of electricity at the present day. 

To Volta is ascribed the invention of the electrophorus, electro- 
scope, and the condenser. The practical unit of potential difference 
is called the ro/^ in his honor. 



ELECTROSTATICS 



353 



AL 



t t + + 



31 



ebonite or a shallow metal dish B^ Fig. 288, about 25 cm. in 
diameter, into which has been poured melted resin or 
shellac, and a flat circular metal disk A^ somewhat smaller 
than the plate and having well-rounded edges. The metal 
disk is provided with an insulating handle of ebonite. 

Let the plate of an electrophorus be stroked with cat's fur and its 
electrification tested. Place the metal disk upon the plate and test 
the kind of electricity that can be taken from its upper surface. 
Touch the disk with the finger in order to "ground" or "earth" it, 
lift it by the insulating handle, and test its charge. Repeat the ex- 
periment, without " grounding " the disk. Explain the result. 

The action of the electrophorus is as follows : When 
the non-conducting plate B is stroked with fur, it is given 
a negative charge. If, now, the metal 
disk A be placed upon the plate, the 
negative on the plate induces a posi- 
tive charge on the lower surface of the 
disk and repels a negative charge to 
the upper side, as shown in Fig. 289. 
When the disk is touched with the 
finger, the negative charge escapes, 
leaving the positive charge, which is 
distributed over the disk when it is 
lifted. Compare § 355. Any number 
of charges may be obtained from the electrophorus with- 
out producing any appreciable change in the charge on 
the plate. 

370. Source of the Energy Derived from the Electrophorus. 
— Although any number of charges can be produced by 
the electrophorus, we know that the energy obtained in 
this manner cannot be brought into existence without the 
expenditure of a like quantity on the part of some work- 
ing agent (§ 64). The source of the energy is obvious 

from the following : 
24 



CZZ 



t)A 



> 



Fig. 289. — Represent- 
ing the Theory of 
the Electrophorus. 



354 



A HIGH SCHOOL COURSE IN PHYSICS 



After the metal disk of an electrophorus has been placed on the 
electrified plate and touched on its upper surface to draw away 
the negative charge (§ 369), connect a gold-leaf electroscope with the 
disk by means of a slender wire. No divergence in the leaves is pro- 
duced, although we know that the disk has a positive charge induced 
by the negative charge on the plate. Slowly lift the disk by the 
insulating handle and the leaves will diverge widely. 

When the disk with its induced positive charge rests 
upon the charged plate, it is at zero potential because it 
has been in connection with the earth. The disk, like a 
weight lying on the ground, manifests no energy until its 
positive charge is separated from the negative charge on 
the plate in opposition to their attractive force. The 
agent, therefore, that lifts the disk from the plate furnishes 
the energy which appears in the disk as electrical energy. 
When the disk is discharged, the electrical energy is 
transformed into heat which appears in the spark that is 
produced. The heat of the spark can be utilized to light 
illuminating gas or explode powder. 

371. The Toeppler-Holtz Influence Machine. — The dis- 
covery of X-rays and their necessity for generators capa- 



Qf^ 




Fig. 290. — The Toeppler-Holtz Electrical Machiue. 



ELECTROSTATICS 355 

ble of producing a continuous supply of electricity has 
brought such generators into extensive use. The simplest 
form of induction, or influence, machines is the Toeppler- 
Holtz^ shown in Fig. 290. This machine makes use of a 
glass plate about 50 cm. in diameter, which is provided 
with six or eight metallic disks, as shown. This plate 
revolves in front of a second stationary plate of glass P, 
which is furnished with two strips of tin foil 6, h\ covered 
with paper sectors called arma- 
tures, or inductors. In front 
of the revolving plate is the 
stationary neutralizing bar B^ 
provided with points and tinsel 
brushes at the ends. (7, 0' are 
two conductors having at their 
other extremes points that come 

close to the revolving plate. , 

The action of the machine is ^^^^ 291. -Diagram Showing 

best understood from a study of Action of the Toeppler-Holtz 

the diagram shown in Fig. 291. 

Imagine a small positive charge placed on the sector h. 
This charge acts inductively through the glass on the rod 
BB\ attracting a negative charge to B and repelling a 
positive one to B^ . These induced charges rapidly escape 
from the points (§ 358) and electrify the glass plate as well 
as each metallic disk as it passes. As the plate rotates in 
the direction shown by the arrow, the disks at the top 
carry negative charges to the right, while those at the bot- 
tom carry positive charges to the left. These disks touch 
the small brushes c and c' ^ through which negative charges 
are given to sector h' and positive to h. The charges on the 
sectors are thus continually increased, an effect which con- 
tinually increases the first inductive action through the 
glass on BB\ Again, when the negatively charged glass 




356 A HIGH SCHOOL COURSE IN PHYSICS 

plate reaches 0\ positive electricity is drawn from the 
points and negative left on the ball 0' . Similarly, on 
the opposite side of the machine, the positively electrified 
plate neutralizes the negative charge that it induces at 
the points C, and thus leaves a positive charge on ball 0. 
The unlike charges on and O continue to increase until 
the difference of potential is sufficient to force a dis- 
charge to take place through the air between them. If 
the distance is not too great, a stream of sparks will appear 
to pass without interruption. The energy transformed in 
each spark is greatly increased by the presence of the two 
Leyden jars shown in Fig. 290. 

EXERCISES 

1. Why is the metal disk of the electrophoriis not charged nega- 
tively by contact with the negatively charged plate? 

Suggestion. — Consider the nature of the plate and the fact that it 
touches the disk in only a very few places. 

2. Explain why an influence machine turns with greater difficulty 
when it is developing a high potential between the balls O and O'. 

Suggestion. — Consider the nature of the electrical forces existing 
between the stationary and moving charges. 

3. Small Leyden jars (see Fig. 290) used in connection with the 
balls of the influence machine cause the production of much brighter 
sparks than would be produced without them. Explain. 

4. With the Leyden jars removed, would the frequency with which 

sparks pass between O and 0' be increased or decreased ? 
< 

SUMMARY 

1. Many substances, as glass, ebonite, sealing wax, etc., 
when rubbed with silk, flannel, fur, etc., possess the prop- 
erty of attracting light objects, and are therefore said to 
be electrified (§ 343). 

2. Electrical charges are of two kinds, called positive 
and negative charges (§ 344). 



ELECTROSTATICS 357 

3. Similar charges repel each other and dissimilar 
charges attract (§ 345). 

4. The electroscope is used to determine the presence 
of a charge and its kind (§ 346). 

5. Equal amounts of positive and negative electricity 
are always developed simultaneously (§ 348). 

6. Charges of electricity are transferred by conductors^ 
as metals, carbon, etc. A substance that will not act as 
a conductor is called an irisulafor; such are dry air, glass, 
mica, rubber, shellac, etc. (§§ 349 and 350). 

7. An electric field is the space around a charged body 
within which neighboring bodies are affected by the 
charge (§ 351). 

8. The electrical effect of an electrified body upon a 
neighboring conductor insulated from it is the result of 
electrostatic induction (§ 352). 

9. The effect of induction by a given charge is to cause 
a dissimilar kind of electricity to appear on the nearer 
side of an insulated conductor and the similar kind on the 
remote side (§§ 353 to 355). 

10. Charges of electricity distribute themselves over 
the exterior surfaces of conductors. The relative quan- 
tities per square centimeter depend on the curvature of 
the surface, being greatest at places of the greatest curva- 
ture, as at points (§ 356 and 357). 

11. Charges are rapidly conveyed away from sharp 
points by air particles, which become charged by contact 
and are then repelled (§ 358). 

12. Charges flow along conductors from places of higher 
to places of lower potential. The potential of the earth is 
regarded as zero (§ 360 and 361). 

13. The union of positive and negative charges of the 
same size produces a cancellation of both (§ 362). 



358 A HIGH SCHOOL COURSE IN PHYSICS 

14. Unit charges are such equal quantities of electricity 
as exert upon each other a force of one dyne when the 
distance between them in air is one centimeter (§ 363). 

15. Electrostatic eapaeity is measured by the number of 
units of electricity required to produce a given potential. 
The capacities of two insulated conductors are proportional 
to the number of units required to bring them to the same 
potential (§ 364). 

16. The capacity of a condenser is made very large by 
bringing the conductor to be charged near another which 
is connected with the earth, but thoroughly insulated from 
the former. The most common form of condenser is the 
Leyden jar (§§ 365 to 368). 

17. The process of induction is employed in the genera- 
tion of large electrostatic charges. See the electrophorus 
(§ 369) and the Toeppler-Holtz machine (§ 371). 



CHAPTER XVII 

MAGNETISM 

1. MAGNETS AND THEIR MUTUAL ACTION 

372. Production of a Magnet. — Magnets present at least 
two properties that are familiar to every one: (1) the ends 
of a magnet will pick up small pieces of iron, as iron filings, 
etc., and (2) a magnet will take a north-and-south posi- 
tion when properly suspended. The following experiment 
will serve to show that an intimate relation exists between 
electricity and magnetism : 

Break off several pieces of watch spring, and balance each of them 
on the head of a pin, as shown in Fig. 292. Observe that they will 
remain indefinitely in any position and will not 
pick up iron filings. Now wrap the pieces of 
spring in a small sheet of paper and place them 
within a helix, or spiral, made by winding ten 
or twelve turns of insulated copper wire upon Fig. 292. — Piece of 
a lead pencil. Discharge a Leyden jar through Watch Spring Bal- 
the helix of wire and then remove the pieces of 

spring. They will be found able to pick up iron filings and small sew- 
ing needles ; and, when balanced on the head of a pin, each piece will 
become stationary only when in a north-and-south position. 

The experiment shows (1) that a moving charge of 
electricity which flows in a helix around a piece of steel 
magnetizes it, and (2) that a magnetized bar of steel re- 
tains at least a portion of the magnetism produced by the 
electric flow. 

373. Natural Magnets or Lodestones. — In ancient times 
iron was mined on some of the islands of the Mediterranean 
and along the coasts of the ^gean sea. It was early 

359 




360 



A HIGH SCHOOL COURSE IN PHYSICS 




observed that an occasional piece of the ore, magnetite 

(chemical symbol, FcgO^), possessed the power of attracting 

small pieces of iron and also 
imparted this property by 
contact to pieces of iron and 
steel. According to some 
writers, the word " magnet " 
is derived from Magnesia in 
Asia Minor, a province in 
which magnetic iron ore is es- 
pecially abundant. Speci- 
mens of ore that possess the 
properties of magnets are 
called lodestones^ or natural 
magnets. See Fig. 293. 

374. Poles of a Magnet.— 
If a magnet be placed on a 
sheet of paper and covered 

with iron filings, it will be found on lifting the magnet 

that the filings cling to the ends in great tufts but leave 

it bare in the middle, as 

shown in Fig. 294. The 

centers of attraction near 

the ends are called the 

poles of the magnet. The 

pole that points toward 

the north when the bar is suspended is the north-seeking, 

or N-pole, and the other the south-seeking, or S-pole. 

375. Law of Magnetic Poles. — Balance upon a pinhead each of 
the magnetized pieces of steel used in § 372 and mark the N-poles 
with small labels. Now bring the N-pole of one piece near the N-pole 
of a suspended one and observe the effect. Present the N-pole of the 
former to the S-pole of the latter. In every case that can be tested it 
will be found that an N-pole repels another N-pole and attracts an 
S-pole. Likewise S-poles repel each other and attract N-poles. 



Fig. 293. — A Natural Magnet. 




Fig. 294. — Showing the Polarity 
of a Magnet. 



MAGNETISM 



361 



The general law of pole action is made clear by this 
experiment, viz. like poles repel each other and unlike poles 
attract. 

The force of attraction or repulsion between two poles is 
inversely proportional to the square of the dista7ice between 
them, i.e. doubling the distance between two poles divides 
the force by four, tripling the distance divides the force 
by nine, etc. Compare with gravitation, § 67. 

376. Artificial Magnets. — The fact that artificial mag- 
nets may be made of any desired form is of great practical 
value. The commonest forms are 
the straight bar magnet and the 
horseshoe magnet shown in Fig. 295. 
These can be produced in any size 
from the small toy magnet up to 
large ones capable of lifting an iron weight of several 
pounds. 

Draw the N-pole of a magnet along a steel nail, from the head 
toward the point. Present the head of the nail to the N-pole of a 
suspended magnet. The observed repulsion shows the head to be 
an N-pole. Also test the polarity of the point of the nail. Repeat 

the processes, using the S-pole of the 
5 first magnet, and ascertain the poles 
produced in the nail. 




Fig. 295. — A Horseshoe 
Magnet. 




N 5 

Fig. 296. — Magnetizing a 
Bar of Steel. 



In every case it will be found 
that when a bar is magnetized by 
contact with one of the poles of 
a magnet, a pole of the opposite 
name is formed at the point last touched by the magnet, 
as illustrated in Fig. 296. 

377. Magnetic Substances. — Practically only iron and 
steel are affected by a magnet, although the substances 
nickel and cobalt are slightly attracted. Bismuth, anti- 
mony, and some other substances are appreciably repelled 



362 A HIGH SCHOOL COURSE IN PHYSICS 

when placed near a strong magnetic pole. Bodies belong- 
ing to the former class are called paramagnetic or simply 
magnetic, substances ; and those of the latter, diamagnetic i| 
substances. 

378. Magnetic Induction. — Let a strong magnet support 
an iron nail by its bead. Test the polarity of the point of the nail. 

Let the point of the first nail support a second 
one by its head and test the polarity of this 
one also. If the nails are not too large, a 
chain of them may be formed as in Fig. 297. 
Now carefully remove the magnet from the 
first nail, and all will fall apart. Repeat the 
experiment after placing a piece of paper be- 
tween the magnet and the first nail. Ascer- 
tain whether absolute contact between the 
Fig. 297. — Each magnet and the nail is necessary in oraer to 

Nail Becomes enable the first nail to support the second. 

a agnet. rpj^^ action of the magnet on the nail is 

simply weakened by the intervention of the 
paper. 

A piece of iron or steel becomes a magnet by induction 
when brought in contact with^ or close to^ a magnetic pole. 
The effect takes place through all substances except large 
masses of iron or steel. When the polarity of the nails 
is tested, it is found that the N-pole of the magnet, for 
example, produces an S-pole on the near end of the nails 
and an N-pole at the remote end. When the magnet is 
removed, it will be found that a portion of the induced 
magnetism is retained by the nails, 

379. Retentivity of Magnetism. — It has been evident 
in many of the preceding experiments (§§ 372, 376, 378) 
that a magnetized piece of steel loses only a part of its 
magnetism when it is removed from the magnetizing in- 
fluence. It is almost impossible to find a piece of iron 
that will not retain a little magnetism after being brought 
in contact with a magnet, although the amount retained is 




MAGNETISM 



363 



often very sliglit. The property of retaining magnetism is 
called retentivity. Hardened steel possesses the property 
of retentivity in a high degree, but in soft iron the reten- 
tivity is very small. 

380. Magnetic Fields. — A magnetic field is a region in 
which magnetic substances experience magnetic forces. A 
magnetic field may be represented in the same manner as 
an electric field (§ 351). Each magnetic line of force 
shows at every point in it the direction of the force at that 
point. Magnetic fields are easily mapped by the help 
of iron filings, as in the following experiment : 

Let a bar magnet be covered with a sheet of paper and fine iron 
filings strewn over its surface. If the paper is slightly jarred by tap- 
ping the table on which the magnet rests, the filings will arrange 
themselves in chains stretching from pole to pole. Tliese chains 
show the direction of the magnetic forces at the points through which 
they pass. 

The direction of the lines of force in the field around a 
short bar magnet is shown in Fig. 298. Each particle 
becomes a magnet by induction (§ 378) and turns until it 
lies lengthwise in a line of force. The poles of one parti- 
cle attract the opposite poles of the neighboring particles, 
and thus the filings unite and form chains which extend 
from pole to pole. 
The direction of a line 
of force is assumed to 
be from an N-pole to 
an S-pole through the 
air. The strong parts 
of tlie field are the re- 
gions near the poles, 
as shown by the 
heavy, distinct lines „ ^,,^ ,„, ,^ ^. „. , , , 

•^ Fig. 298. — The Magnetic Field around a 

of filings. Bar Maguet. 




364 A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 



299. — Showing the Magnetic Field 
between Unlike Poles. 



Figure 299 was 
made by placing two 
bar magnets side by 
side with their unlike 
poles pointing in the 
same direction. It 
is at once evident that 
an intense field of 
force is produced be- 
tween unlike poles 
when placed near 
each other. This fact 
is utilized in some of 
the practical applica- 
tions of magnetism. 

Figure 300 shows 
the result obtained by 
placing two short bar 
magnets parallel, but 
with their like poles 
turned in the same 
direction. A comparison of this with the figure just pre- 
ceding shoAvs at once the presence of a weak field of force 
between similar poles. In fact, the re- 
pellent action between the lines of force 
is obvious. No lines of force from one 
pole enter another pole of the same 
name. Since the poles are alike, it is . 
plain that this field represents the case 
of repulsion, while Fig. 299 shows the 
condition for attraction. 

381. Magnetic Permeability. — Mag- 
netic lines of force find an easier path ^ig- ^oi.-a Magnetic 

. Field Distorted by a 

through iron or steel than through air. pfece of n-ou A. 




Fig. 



200. — Showing the Magnetic Field 
between Like Poles. 




MAGNETISM 365 

When a piece of iron, A, Fig. 301, is placed in a magnetic 
field, lines of force bend from their original course in 
order to pass through the iron. Thus lines of force are 
sent through the metal. The relative ease with which mag- 
netic lines of force pass through a substance is called its mag- 
netic permeability. The permeability of air is regarded as 
unity. 

EXERCISES 

1. What would be a suitable substance for permanent artificial 
magnets ? 

2. It is desirable in a certain instrument to use a substance that is 
easily magnetized, but which will lose its magnetism when the mag- 
netizing influence is removed. AVhat would be the proper substance ? 

3. Each of two nails hangs fr^lfei the N-pole of a permanent mag- 
net. Will the ends of the nails remote from the magnet attract or 
repel each other ? 

4. Four nails are placed lengthwise in a row without touching each 
other. Draw a figure showing the magnetic fields produced when the 
S-pole of a magnet is placed near one end of the row. 

5. The N-pole of a magnet is held near a point on the circumfer- 
ence of an iron ring. Show by a diagram the position of the induced 
magnetic poles. 

2. MAGNETISM A MOLECULAR PHENOMENON 

382. Demagnetization by Heating. —Magnetize a piece of 

watch spring and note the quantity of iron filings that it will support. 
Heat the spring red-hot and test its magnetism again. It will be found 
to have lost its power of picking up filings as well as repelling either 
pole of a suspended magnet. 

We have already learned in § 213 that heating a body 
simply increases its molecular motion. This experiment 
shows, therefore, that when the molecular motion reaches 
a certain degree, practically all magnetism is destroyed. 

383. Demagnetization by Molecular Rearrangement. — 

Bend a piece of iron wire in the form shown in Fig. 302. Magnetize 
it by stroking it several times with a strong magnet. Test its power 



366 A HIGH SCHOOL COURSE IN PHYSICS 

to pick up filings and to repel the pole of a suspended magnet. Now 
grasp the ends with pliers and give the wire a vigorous twist. If the 
wire is now tested as before, it will be found 
^ to have lost its magnetism. 

Fig. 302.— Wire may 

be Demagnetized by It is obvioUS that the effect of twist- 

^^^ ^°^' ing is to give all parts of the wire a new 

molecuhir arrangement. Accompanying this disarrange- 
ment is the disappearance of the magnetism, as shown by 
the experiment. 

384. Magnetism Not Simply at the Poles. — Magnetize a 

piece of watch spring about 5 in. long and test the location of its poles 
by presenting it to a suspended magnet. Break it at the center and 
test the pieces. It will be found j^at each piece is a perfect magnet, 
for two new poles will have developed at the point which was at first 
neutral. Break one of the pieces at its center and again each piece 
will be a perfect magnet. 

Let the original magnet be represented by (1), Fig. 303. 
The polarity manifests itself only at the ends. When the 
mascnet is broken, as „ „ 

at P, the condition 0M- 'mM ( / ) 

shown in (2) is the n s n s 

£^f:!" refill 0^': ':''4% ( 2 ) 

result. Again, on ^^ -^^^ te^ ^^ 

breaking the two f i j r=l > f C \ 1 (.9^ 

p a r t S , t h e result ^^^ 3^3 _ ^^^^^ ^^ Breaking a Magnet. 

shown in (3) is ob- 
tained. It may be imagined that this process be carried on 
• even to the separation of ultimate particles, i.e. the mole- 
cules. There can be no doubt that each molecule would 
prove to be a magnet having two poles. 

385. Theory of Magnetization. — The results obtained 
in the experiments of the preceding sections lead to the 
theory of magnetization which assumes that every molecule 
of iron or steel is a magnet even when the bar of which it is a 
part is not magnetized. Magnetization consists in causing 



MAGNETISM 



367 



^nHO*^ j£^C3!M' 



>iiir-i % %:> ' 



■^^^.i^llllO 



magnetized Bar of Iron or Steel. 






Fig. 305. — Illustrating the Condition in a 
Magnetized Bar. 



the molecules to arrange themselves in a certain order. 
In an unmagnetized bar of iron the condition is repre- 
sented as in Fig. 304. Here each of the little rectangles 
represents a pivoted magnet, the N-pole being shaded. It 
will be seen that the small magnets arrange themselves in 
small groups so that 
unlike poles neu- 
tralize each other 
throughout the bar. 

The bar, therefore, Fig. 304.-— illustrating the Condition in an Uu 

manifests no polar- 
ity. 

When a magnet- 
izing influence is 
brought to bear 
upon the bar of iron, 
the molecules swing 
round until a condi- 
tion approximating 
that shown in Fig. 
305 results. Since 
the general direction 
of the N-poles is to- 
ward the left and the S-poles toward the right, the ends of 
the bar will show polarity. At a short distance from the 
ends and throughout the middle of the bar the proximity 
of unlike poles brings about a neutralization of polarity in 
these places. A jar serves to break up this artificial ar- 
rangement, which thereby destroys the magnetism, and the 
condition shown in Fig. 304 is resumed. 

386. A Saturated Magnet. — If the theory of magnetiza- 
tion just described is correct, it will be found impossible 
to magnetize a piece of iron beyond the limit reached 
when all the molecules have been turned as represented 



IMP MP IIP mo Qnn mnn nio imn iniri nnn inio iMn 1^ 
flip ME I11E nnn imn [inn imn innn fnmn innn imri n^ 
(DP mo ffliD QiiD cnp imp nno inin mn iniri iiirri 1^ 
QiiD DHD iia !!□ EDn QD inn DIO IID Dn 0^ 



Fig. 306. 



- Illustrating the Condition in a 
Saturated Magnet. 



368 A HIGH SCHOOL COURSE IN PHYSICS 

in Fig. 306. Tliis has been found experimentally to be 
the case. In this condition a magnet is said to be satu- 
rated. Hence, a saturated magnet is one upon which an 
increase of the magnetizing influence has no effect. 

EXERCISES 

1. How would you determine the poles of a magnet? Give two 
methods. 

2. Why is a permanent magnet injured when dropped? 

3. If iron be heated and the)i cooled in a magnetic field, it will be 
found to be magnetized. Explain. 

4. Explain how jarring a bar of steel will aid in magnetizing it, 
but jarring a magnetized piece of steel will weaken its magnetism. 

3. TERRESTRIAL MAGNETISM 

387. The Compass. — One of the earliest properties dis- 
covered regarding a magnet is its tendency to take a defi- 
nite position when placed on a pivot or suspended. A 
magnet that is so pivoted as to turn freely in a horizontal 
plane is called a compass. The invention of the compass 
is attributed to the Chinese. 

The first satisfactory explanation of the action of the 
compass was given by Gilbert ^ about 1600 a.d. The 

1 William Gilbert (1540-1603). The renown of Gilbert rests largely 
upon the fact that he was one of the first to recognize the value of experi- 
mentation and also upon his great work entitled De Magnete, which was 
published in London in 1600. His most celebrated experiments were 
made with magnets and magnetic bodies, and his results and conclusions 
are contained in the book just named. Gilbert made the discovery that 
the earth is a great magnet and demonstrated this by constructing a 
small sphere of lodestone. With this "terrella," or "little earth," he 
was not only able to show why magnets point to the north, but he also 
explained the declination and inclination of magnetic needles. 

Gilbert also made important discoveries in the subject of Electricity. 
He was probably the first to clearly recognize a distinct difference between 
magnetized and electrified bodies. He showed that many substances be- 
sides amber could be electrified by rubbing ; e.g. glass, resins, sealing wax, 



MAGNETISM 369 

assumption is made that the earth is a great magnet, but 
the reason for its being one still remains a mystery. 
However, we are sure that the earth is surrounded by a 
magnetic field, and, in proof of this fact, the following 
experiment can easily be made. 

Hold a bar of iron or a slender gas pipe about 2 feet long in a north- 
and-south plane, tilting the north end down 60° or more. Now give 
the bar several vigorous taps with a stone and then test the ends for 
polarity by presenting them to the poles of a suspended magnet. The 
end of the bar toward the north will be an N-pole, and the other an 
S-pole. Reverse the bar and repeat the processes. The polarity of 
the bar will be found reversed. 

This experiment makes use of the earth's magnetic field 
in the magnetization of the iron bar. The jarring is ef- 
fective only in lending assistance to the rearrangement of 
the molecules under the inductive influence of the earth's 
field (§§ 380, 385). The compass needle simply behaves 
as any small magnet would in a magnetic field, pointing 
in a general north-and-south direction because of the 
influence of the earth's lines of force. 

388. Declination of the Needle. — Suspend a magnetized knit- 
ting needle on a fiber taken from the cocoon of a silkworm, or on an 
untwisted filament of silk floss. Stretch a cord below the needle pre- 
cisely in a north-and-south line (given by the shadow of a plumb line 
or a vertical window frame at noon, sun time). The experiment 
should not be made within several feet of an iron pipe or beam, nor 
within a foot of steel nails. It will be observed that the needle does 
not take a position parallel to the north-and-south line except in 
certain localities. 

etc. These he named " electrics," from the Greek word electron^ meaning 
amber. He was also the first to use the word " electricity." 

Gilbert was educated in medicine at Cambridge, England, and was 
appointed court physician by Queen Elizabeth. At the death of the queen 
in 1603, he was reappointed by her successor, James I, but his death 
occurred in November of that year. 



370 



A HIGH SCHOOL COURSE IN PHYSICS 



The earth's magnetic lines do not coincide with the geo- 
graphical meridians ; consequently the magnetic needle, 
directed by the earth's field, will not point geographically 
north. This fact was known as early as the eleventh 
century ; but it was first discovered by Columbus, on his 
memorable voyage of 1492, that the direction indicated 
by the compass changes as one passes from place to place 
over the earth's surface. The angle between the direction 
of the needle and the geographical meridian is the declina- 
tion of the needle. 

Lines that are so drawn upon a map as to pass through 
places at which the declination is the same are called 




Fig. 307. — Map Showiug Magnetic Declinations. Lines of no Declination 
are Marked "0." The Numbers Show the Declination in Degrees and the 
Letters Show W^hether it is East or West. 



isogenic lines. Figure 307 shows approximately the mag- 
netic declination at all places on the surface of the earth. 
The heavy line, called the agonic line, shows the regions 
where the needle points due north. At all points in the 
United States and Canada lying east of the agonic line, 



MAGNETISM 



371 



the declination is towards the west ; for all points west 
of this line, the declination is east. At the present time 
(1910) the agonic line passes a little west of Lansing, Mich., 
and slightly east of Cincinnati, Ohio, and Charleston, S.C. 
It is moving very slowly westward. 

389. Inclination of the Magnetic Needle. — Thrust an unmag- 

netized knitting needle through a cork, and close to and at right angles 
with it pass a straight, slender sewing needle. Cut away a portion 
of the cork until the system will balance in 
any position on the edges of two tumblers 
when the longer needle is placed east-and- 
west. Magnetize the knitting needle by strok- 
ing one end with the N-pole of a magnet and 
the opposite end with the S-pole. Now place 
the system in a north-and-south position on 
the tumblers. The N-pole of the needle will 
appear heavier than the S-pole and will dip 
until the angle between the needle and the 
horizontal plane is about 70°. (See Fig. 308.) 

The balanced, or dipping, needle 
simply places itself parallel with the 
lines of force of the earth. In the Northern States these 
lines are inclined about 70° from the horizontal. The 
angle between the earth's magnetic lines of force and a hori- 
zontal plane is called the inclination, or dip, of the needle. 
The angle of inclination increases as one approaches the 
magnetic poles of the earth, where it is 90°. Near the 
geographical equator the inclination is about 0°, and in 
the southern hemisphere the needle inclines with its S-pole 
below the horizontal plane passing through its axis. 




Fig. 308. — The N-pole 
of a Balanced Needle 
Tends to Dip Down. 



EXERCISES 

1. Account for the fact that the iron beams of a building, gas and 
water pipes, and other bars of iron are usually magnetized. 

2. How would a compass behave while being carried entirely 
around the earth's magnetic pole along the Arctic Circle ? 



372 A HIGH SCHOOL COURSE IN PHYSICS 

3. How would a dipping needle be of assistance in locating the 
magnetic poles of the earth ? 

4. Account for the fact that a surveyor's compass needle is often 
provided with a small adjustable weight for balancing. 

5. Place a ruler upon the table, giving it approximately the declina- 
tion of the needle at New York ; at Los Angeles ; at Iceland. 

6. Ascertain by experiment whether a floating magnet tends to 
drift toward the north. Account for the result. 

Suggestion. — Consider whether the attraction of one of the 
earth's poles for a pole of a magnet is greater than its repulsion for 
the opposite pole of the same magnet. 

7. The upper end of a pipe driven into the ground was found to be 
an S-pole. Explain. Suggest an experiment by which your explana- 
tion can be tested. 

SUMMARY 

1. A bar of steel may be made a magnet by discharging 
a Leydeii jar through a helix of wire wound around it 
(§ 372). 

2. Magnets also appear in nature in an ore of iron 
called magnetite. Specimens of this ore that are magnets 
are called natural magnets (§ 373). 

3. On account of the tendency of a magnet to place 
itself in a north-and-south position, one end is called the 
north-seeking, or N-pole ; the other, the south-seeking, or 
S-pole (§ 374). 

4. Poles of the same name repel each other, and 
those of unlike name attract (§ 375). 

5. Artificial magnets may be made by stroking bars 
of steel with a magnetized body (§ 376). 

6. Substances that are attracted by a magnet are called 
paramagnetic (or simply magnetic') substances ; those that 
are repelled, diamagnetic. Iron is the most magnetic of 
all substances (§ 377). 

7. A piece of iron or steel becomes a magnet by indue- 



MAGNETISM 373 

tion when brought near the pole of a magnet. An N-pole 
always induces an S-pole on the nearer end of a neighbor- 
ing bar of iron or steel, and an S-pole induces an N-pole 
(§ 378). 

8. The region around a magnet in which magnetic 
substances experience magnetic forces is called a magnetic 
field (§ 380). 

9. Magnetic lines of force pass more readily through 
iron than air. Different grades of iron and steel differ 
also in the ease with which lines of force pass through 
them. They are therefore said to differ in permeability 
(§ 381). 

10. The magnetism which would ordinarily be retained 
by iron or steel may be reduced or entirely destroyed by 
heating, jarring, twisting, etc. Hence magnetism is con- 
sidered to be a molecular phenomenon. The molecules of 
iron and steel are supposed to be small magnets having two 
poles. Magnetization consists in giving the molecules a 
new arrangement in which the N-poles point in general in 
one direction and the S-poles in the other (§§ 382 to 386). 

11. A magnet so pivoted as to turn freely in a horizontal 
plane is called a compass. It takes a general north-and- 
south position on account of the earth's magnetism (§ 387). 

12. The angle that measures the deviation of the compass 
from the geographical meridian is called the declination of 
the needle. This angle varies greatly with different lo- 
calities. The agonic line is the line passing through points 
where the declination is zero. It is now moving slowly 
westward (§ 388). 

13. The angle between the earth's magnetic lines and a 
horizontal plane is called the inclination^ or dip^ of the 
needle. The dip varies from zero at the magnetic equator 
to 90° at the earth's magnetic poles (§ 389). 



CHAPTER XVIII 



VOLTAIC ELECTRICITY 



1. PRODUCTION OF A CURRENT — VOLTAIC CELLS 

390. Maintaining a Continuous Discharge of Electricity. 

— It was shown in § 372 that the moving charge of elec- 
tricity obtained when a Leyden jar is discharged through 
a helix of wire is able to magnetize pieces of steel. But 
electricity would be of very little service if we were obliged 
to depend upon the momentary flow occasioned by such a 
discharge. The great practical value of electricity as a 
working agent lies in the fact that it is possible by differ- 
ent means, now to be studied, to maintain a continuous flow 
of electricity, or in other words, a steady current. The 
following experiments afford an opportunity to compare 
these two kinds of discharge, — viz. the momentary and 
the cojitinuous. 

1. Place one end of a coil of insulated wire within which are several 
pieces of soft annealed iron wire near one of the poles of a light 

pivoted magnetic needle as 
shown in Fig. 309. Discharge 
a highly charged Leyden jar 
through the wire of the coil 
and observe the effect upon 
the needle. The pole will be 
either attracted or repelled. 
Except for the effect of the 
magnetism remaining in the 
iron, the action is only tempo- 

FiG. 309. — Magnetizing Iron by a Mo- ^«^y- Recharge the jar and 
mentary Discharge of Electricity. discharge it through the coil 

374 




VOLTAIC ELECTRICITY 375 

in the opposite direction. If the first discharge produced repulsion, 
this one will set up an attraction. 

2. Attach one end of the coil shown in Fig. 310 to a strip of copper 
about 10 cm. long and the other to a strip of zinc. Now dip both 
plates in a very dilute solu- ^_-^,-^~-,^__^ 

tion (about 1 : 40, by volume) ^r ^"""v-^^,^ 

of sulphuric acid, keeping A \ 

the plates from touching ^ teflffCflfliSl ^''^TV^'^'lir''^'''^ 

each other. One of the mag- ^/w h ¥4^=^=========^l 

netic poles of the pivoted /^ |OpF^^^fW,H 

needle will swing round ^> |l-\ l|]]]]|t l/i 

toward the coil and will not ili ¥" IvM 

resume its original position ^^f^^^'^'^^^^^w 

so long as the plates remain ^"'' ^ ' — 

in the liquid. Reverse the Fig. 310.— Magnetizing Iron by a Continuous 

connections at the ends of 

the coil, and the opposite end of the needle will be attracted. 

These experiments show a marked similarity between 
the effect produced by the discharge of a Leyden jar and 
that produced by the plates of zinc and copper suspended 
in a solution' of sulphuric acid. The iron is magnetized 
in both instances, and the magnetism is even reversed by 
reversing the connections. But since the effect in Ex- 
periment 2 lasts as long as both plates remain in the 
liquid, it is clear that a continuous discharge can be main- 
tained by the means employed. A continuous discharge^ 
or movement^ of electricity is called an electric current. 

391. The Voltaic Cell. — The combination of zinc, cop- 
per, and dilute sulphuric acid used in Experiment 2 of the 
preceding section constitutes a voltaic cell. This method 
of producing a current was first used by Volta ^ in 
the year 1800 ; hence the name. There are many kinds 
of voltaic cells, differing from one another in respect to 
the materials used in their construction. Some of the 
most important ones are treated later on. 

1 See portrait facing page 352. 



376 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 311, — Charging an Electroscope 
from Voltaic Cells. 



392. Electrical Charges produced by Voltaic Cells. — 

Provide two perfectly flat metal disks about 3 in. in diameter. Attach 
one disk to the top of a sensitive gold-leaf electroscope and give it a thin 
coating of shellac. To the second disk attach an insulated handle 
and place it upon the first. The two plates thus form a condenser of 
large capacity. Why? See §365. Form a series^ of voltaic cells 
(discarded dry cells will usually do) and touch the wire leading from 

the zinc plate to the upper disk Aj 
Fig. 311, and the wire leading from 
the carbon (or copper) to disk B. 
Remove the wires and lift the 
upper disk. The leaves of the 
electroscope will diverge. On bring- 
ing an electrified stick of sealing 
wax near the instrument, the diver- 
gence will be decreased, thus show- 
ing that the carbon (or copper) 
plate conveyed to the lower disk a 
positive charge. By repeating the 
experiment with the wires reversed, it will be found that a negative 
charge can be taken from the zinc of the cell. 

It becomes clear from this experiment that the terminals 
of a voltaic cell and the ivires leading from them are electri- 
cally charged^ — a positive charge being home by the carbon 
(^or copper^ and a negative charge by the zinc. Hence the 
copper of a cell is called the positive pole^ and the zinc, the 
negative pole. The voltaic cell, therefore, affords a case of 
two oppositely electrified bodies, which on being joined by 
a conductor set up an electrical discharge. Furthermore, 
this discharge, or current, is continuous ; for, as fast as 
the charges become neutralized, they are renewed by the 
chemical changes taking place within the cell. 

393. Electromotive Force. — As shown in the preceding 
section, a voltaic cell develops a charge of positive electri- 
fication on the copper plate and a charge of negative on 

1 If the disks are perfectly flat and the insulating material between 
them is thin enough, the experiment can be made by using a single cell. 



VOLTAIC ELECTRICITY 377 

the zinc. The difference of potential that the cell maintains 
between these two charges when the plates are not connected by 
a conductor is the electromotive force (abbreviated E. M. F.) 
of the cell. E. M. F. is sometimes called electric pressure 
and is the cause of an electric flow in a circuit. It is not 
a, force in the sense in which that term is used in mechanics, 
since its tendency is to move electricity and not matter. 
The unit of E. M. F. is the volt, which is approximately 
the E. M. F. afforded by a cell containing zinc, copper, and 
dilute sulphuric acid. 

394. An Electrical Circuit. — The entire conducting path 
along which a current of electricity flows is called an elec- 
trical circuit. The circuit comprises not only the wire 
with which the plates are connected, but also the plates 
and the liquid of the cell. When an instrument, for ex- 
ample, is to be introduced into the circuit, it is so connected 
with the plates of the cell as to be traversed by the cur- 
rent and is thus made a part of the circuit. Separating 
the circuit at any point is called opening, or breaking, the 
circuit ; and joining the separated ends is called closing, or 
making, the circuit. When a circuit is broken, no current 
can flow, and the circuit is therefore said to be interrup>ted, 

395. Action of a Voltaic Cell. — l. Place a strip of commer- 
cial zinc in very dilute sulphuric acid (about 1 : 40, by volume) and 
observe the effect. Very small bubbles will be seen to rise from the 
zinc and pass off at the surface of the liquid. These are bubbles of 
hydrogen gas which is liberated from the acid by the chemical action. 
If a small piece of zinc be left in the acid, it will soon dissolve, leaving 
behind only a few small flakes of insoluble impurities. Repeat the 
experiment with the strip of copper. No action will be observed. 

2. Touch the zinc plate just used to a small quantity of mercury. 
Some of the mercury will be found to cling to the plate. With a cloth 
or sponge spread the mercury over the wet surface of the zinc. Now 
on placing the zinc in the acid, no bubbles will be seen. Place the 
copper plate in the acid with the zinc, but not in contact with it, and 
connect the two metals by means of a wire. Bubbles will now be 



378 



A HIGH SCHOOL COURSE IN PHYSICS 



observed rising from the copper plate. It is under these conditions, 
as we have seen in § 390, that a magnetic effect is derived from the 
wire which connects the two plates. If the circuit be now broken at 
any point, the bubbles cease to form. The mercury has thus pre- 
vented the formation of hydrogen bubbles at the zinc plate, which, 
nevertheless, continues to dissolve and decrease in size as long as the 
electrical connection is maintained between it and the copper. 

It is clear from Experiment 1 that the acid used acts 
very unequally upon the two plates of a voltaic cell. 
It is this difference in the chemical action that gives 
rise to the difference of potential between the positive and 
negative charges found in § 392. The greater the disparity 
in the chemical actions at the two plates^ the greater the dif- 
ference of potential maintained hy the cell. Furthermore, 
the experiment shows that the negatively charged plate 
(i.e. the zinc) is dissolved by the acid, while the posi- 
tive copper plate from which hydrogen rises during the 
operation of the cell remains apparently unchanged. 

396. Theory of a Voltaic Cell. — The theory of the simple 
voltaic cell described in § 390 rests upon the hypothesis of Clausius, 

(1822-1888), a German physicist. This 
hypothesis, which is based upon a large 
amount of experimental evidence, 
states that many of the molecules in 
a dilute solution of a substance " split 
up " into two parts called ions. Hence, 
when sulphuric acid (H2S04)i is di- 
luted, two kinds of ions are formed, 
those of hydrogen (H) and those of 
SO4, called the sulphions. See Fig. 
312. Furthermore, the hydrogen ions 
bear positive charges of electricity, while the sulphions carry negative 
charges. As shown in § 395, zinc has a strong tendency to dis- 
solve in dilute sulphuric acid, while copper has not. In the process 

1 This chemical formula expresses the fact that each molecule of 
sulphuric acid is composed of two atoms of hydrogen, one of sulphur, 
and four of oxygen. 



Copper 

i r? 




Fig. 312. — Diagram Showing 
Ions in a Voltaic Cell. 



VOLTAIC ELECTRICITY 



379 



the sulphions in the liquid attack the zinc plate, from which they 
abstract some of the metal to form zinc sulphate (ZnSO^), a white 
substance that dissolves at once in the liquid. In this process the 
negative charge carried by each sulphion concerned in the action is 
given up to the zinc plate, which thus becomes charged with negative 
electricity. Again, for each negatively charged ion that engages with 
zinc, a positive hydrogen ion from the liquid gives up its charge to 
the copper, thus charging it with positive electricity. After the hy- 
drogen ions have discharged their electricity to the copper plate, they 
become free hydrogen, which collects in small bubbles at this plate 
and rises to the surface. 




397. Local Action and its Prevention. — The results of 
Experiment 2 (§ 395) show that a coating of mercury 
upon the zinc plate prevents the formation of hydrogen at 
its surface when it comes in contact with the acid. The 
reason is because pure zinc will not dis- 
solve in pure sulphuric acid ; and the 
mercury dissolves from the plate only the 
pure zinc which is then coated over the 
surface, thus overlaying the impurities 
with an amalgam of zinc. After a time 
these impurities, which are mainly carbon 
and iron, become exposed to the acid, 
and local currents . are set up between 
them and the neighboring portions of the 

plate, as indicated by the arrows in Fig. 313. The gen- 
eration of electric currents between the zinc of a cell and its 
impurities is called local action. Local action is a wasteful 
process, but can obviously be prevented by employing 
pure zinc or by coating an impure zinc plate with mer- 
cury. This treatment is known as the amalgamation of 
the zinc plate. 

398. A Mechanical Analogy. — The mechanical device 
shown in Fig. 314 is of assistance in making clear the 
action of a voltaic cell. Imagine a rotary pump P to be 



Fig. 313. — Wasting 
of the Zinc by 
Local Action. 



380 



A HIGH SCHOOL COURSE IN PHYSICS 



\h 



placed in a U-tube and arranged to be turned by the 
weight TF. When the wheel of the pump is turned, water 

is forced to a greater height in one arm 
than in the other. The wheel, how- 
ever, will come to rest when the back 
pressure due to the difference of level 
on the two sides of it just equals the 
pressure exerted by the wheel. If there 
is no friction, the device will maintain 
the difference in level h as long as no 
water escapes. 

In this system the difference of level 
h maintained by the pump is analogous 
to the difference of potential (E. M. F.) maintained be- 
tween the plates of a voltaic cell. Just as the pump 
ceases to turn when a certain difference of level is reached, 
so the chemical action of a perfect cell stops wdien the 
difference of potential between the positive and negative 
charges on the plates has attained a certain value, which 
will depend on the nature of the ma- 
terials used in its construction. 
399. A Cell in Action. — The 



\w 

Fig. 314. — a Mechan- 
ical Analogy of Cell 
Action. 



1 



W 




case 
of a voltaic cell is modified as soon 
as the circuit is completed and a cur- 
rent allowed to flow ; so also is that 
of the pump and water. Imagine a 
pipe T^ Fig. 315, to connect the two 
arms of the U-tube. On account of 
the difference of water level, the liquid 
will flow through T^ and the wheel will 
continue to turn, since now the back 
pressure will have been reduced. The 
difference of level will decrease to h\ whose value will 
depend on the friction offered to the current flow in T. 



Fig. 315. — When Water 
is Allowed to Flow 
through T, the Wheel 
at P will Continue to 
Turn. 



VOLTAIC ELECTRICITY 381 

Similarly, when the plates of a voltaic cell are connected 
by a conductor, the charge on the positive plate (copper) 
moves toward the negative plate (zinc), and the potential 
difference is diminished. The chemical action now goes 
on vigorously in its attempt to restore the charge that 
has passed through the conductor. Hence the difference 
of potential between the two poles of a cell when a current 
is flowing will be less than that when the circuit is open. 
This value is no longer called the E. M. F. of the cell, but 
is termed the fall of potential, or difference of potential 
between the plates of the cell. 

400. Deflection of a Magnet by a Current. — Hold the wire 

joining the plates of a simple voltaic cell over and parallel to a pivoted 
magnetic needle, Fig. 316, and 

then close the circuit by placing /^ ^ 

the plates in the liquid, or by / ^ lL-^^' /^^^""^^V-- 

means of a key inserted anywhere C/^^~ ~ _2-^— :!x "^^^J^^^^^Z^^c^ Z 
in the circuit. The needle will be [ lU^^ 

turned on its pivot and finally come JL P S - 

to rest at an angle with the con- 
ductor carrying the current. If 
the current be passed in the op- 
posite direction above the needle, the deflection is opposite to the 
first. The wire may now be placed below the needle, and the direction 
of the deflections obtained. If the conductor be placed at the side, 
or near the end, of a suspended magnet, deflections are also obtained. 

This experiment confirms the result previously found 
in § 390, viz. that the region around a wire through which 
an electric charge is moving has magnetic properties. 
This experiment was first performed by Oersted,^ a Danish 
physicist, in 1819. Oersted's discovery is of great his- 
torical interest, since it was the first evidence obtained in 
regard to the magnetic effect of a current of electricity 
and has led to results of the greatest practical importance. 

1 See portrait facing page 382. 



Fig. 316. — a Magnetic Needle is De- 
flected by an Electric Current. 



382 



A HIGH SCHOOL COURSE IN PHYSICS 




fsm- 



FiG. 317. — The Thumb Shows the 
Direction in AVhich the N-Pole 
Moves. 



By observing the direction in which the N-pole of the 
magnetic needle is moved in relation to the direction of 
the electric current, the following rule will be found to 

apply : Let the fingers of the out- 



Pv'**''^^?!^ s^re^(?Ae<i right hand point in the 
direction of the current flow in 
the wire and the palm he turned 
toiuard the needle ; the extended 
thumb will then show the direction 
of the deflection of the N-pole of 
the needle. The rule is of convenience in determining the 
direction of a current when its effect upon a magnetic 
needle is known. The manner of applying this rule is 
made clear by Fig. 317. 

401. The Galvanometer. — The effect discovered by 
Oersted is of great service in the galvanometer, an instru- 
ment used for detecting and measuring electric currents. 
The following experiment will show how the effect of a 
current on a magnetic needle can be so increased as to 
make it possible to discover even very feeble currents. 

Place a compass needle, below which is a graduated circle, within 
a single turn of wire as in (1), Fig. 318, and read the deflection on the 
scale. Now wind the same wire three or four times around the com- 
pass, always keeping the wire parallel to 
the original position of the needle, and 
again read the deflection. The second 
reading will be much greater than the 
first. A current that will scarcely move 
the needle when a single turn of wire is 
used will be found to produce a marked 
effect when tested with the coil of several 
turns. 



The magnetic forces due to the 
electric current in all parts of the 
coil tend to turn the needle in one 




Fig. 318. — Oersted's Effect 
Applied in the Galvanom- 
eter. 



HANS CHRISTIAN OERSTED (1777-1851) 




Oersted's famous experiment of 1819 on 
the deflection of a magnet by an electric 
current was the beginning of the science of 
electro-magnetism. This experiment dem- 
onstrated the long-sought connection be- 
tween electricity and magnetism and served 
to point out the line of experimentation 
that has brought the science up to its pres- 
ent-day development. 

Oersted was born in Langeland, a por- 
tion of Denmark, studied at Copenhagen, 
and afterwards became a professor at the 
university and polytechnic schools of that 
city. 



DOMINIQUE FRANCOIS JEAN ARAGO (1786-1853) 



The year following Oersted's discovery, 
Arago, a noted Parisian astronomer and 
physicist, observed that iron filings cling to 
a conductor carrying an electric current. 
A little later it was shown by Sir Humphry 
Davy of England that the filings arrange 
themselves in magnetized chains around the 
conductor. 

Arago was one of the first advocates of 
the wave theory of light. The beautiful 
tints produced when polarized light passes 
through certain crystals were discovered by 
him in 1811. Arago planned a method for 
measuring directly the velocity of light in 
air and water, but failing eyesight pre- 
vented carrying out his experiments. He lived, however, to see the 
work done by Fizeau and Foucault. 




VOLTAIC ELECTRICITY 383 

particular direction, a fact that becomes clear when the 
rule given in § 400 is applied. Hence by introducing 
a sufficient number of turns of wire and making the 
needle extremely light, a very small current will suffice 
to produce a deflection. 

402. Polarization of a Voltaic Cell. — i. Connect a simple 

zinc and copper cell with a voltmeter ^ (§ 430) or a high-resistance 
galvanometer and read the deflection produced. Short-circuit the cell 
for a short time by means of a wire, thus allowing hydrogen to form 
in large quantities at the copper plate, remove the wire, and again 
read the deflection. It will be less than at first. If the liquid is 
now stirred, the deflection will be increased. Keep the bubbles 
brushed from the copper plate and see if the current can be kept the 
same as at first. 

2. Substitute a carbon plate for the copper and repeat Experi- 
ment 1. (A good carbon plate may be taken from a discarded dry cell.) 
A similar decrease in the current will be observed. While the cell is 
connected with the instrument, pour into the sulphuric acid a small 
quantity of sodium (or potassium) dichromate or chromic acid solu- 
tion. The index of the galvanometer promptly indicates a strong 
increase of electromotive force. The cell may now be short-circuited 
as before, but the E. M. F. quickly resumes its original value after 
the short circuit is removed. 

These experiments show clearly that the collection of 
hydrogen on the copper (or carbon) plate of a cell reduces 
the E. M. F. and, consequently, the current that it sends 
through the conductor. This effect arises from the fact 
that a hydrogen-coated plate now takes the place of the 
copper plate. The diminution of the E. M. F. of a cell 
hy the presence of hydrogen on the copper (or carbon^ plate 

1 For classroom demonstration purposes it is desirable that a volt- 
meter be provided having upwards of 50 ohms resistance and reading 
to about 5 volts. An ammeter having less than an ohm resistance and 
reading to 3 or 4 amperes will also be found of great service. In case 
such instruments are not available, a galvanometer of large resistance (50 
ohms or more) may be substituted for a voltmeter, and one of small 
resistance (less than 1 ohm), for an ammeter. 



384 



A HIGH SCHOOL COURSE IN PHYSICS 



is called the polarization of the cell. If the bubbles be 
removed by stirring, the current will remain near its 
original value. Experiment 2 serves to demonstrate the 
fact that the effect of the hydrogen can be largely overcome 
by chemical means, a method of which advantage is taken 
in many kinds of voltaic cells. In the prevention of 
polarization by chemical action the dichromate acts as a 
depolarizer for removing the hydrogen from the carbon 
plate. This it does by supplying an abundance of oxygen, 
with which the hydrogen unites chemically to form 
water. 

403. The Dichromate, or Grenet, Cell. — In this cell a 
zinc plate Z, Fig. 319, is usually placed between two 

plates of carbon which are joined together 
by metal at the top. The liquid is dilute 
sulphuric acid (HgSO^) to which is added 
dichromate of sodium (or potassium) or 
chromic acid as a depolarizer. See § 402, 
Exp. 2. 

The dichromate cell is capable of giving 
a strong current for a short time and for 
this reason has been largely used in ex- 
perimental work. It has, however, been 
largely replaced by the " dry " cell (§ 407) 
and storage battery on account of their 
greater convenience. One disadvantage of 
the dichromate cell is the necessity of with- 
drawing the zinc from the acid by the rod 
A when the cell is not in use. 

404. The Daniell Cell. —The Daniell cell. Fig. 320, 
consists of a glass jar containing a saturated solution of 
copper sulphate (blue vitriol, CuSO^) in which stands 
a large sheet copper plate C. The copper plate encircles 
a porous cup of unglazed earthenware which contains a 




Fig. 319. — The 
Grenet, or 
Dichromate 
Cell. 



VOLTAIC ELECTRICITY 



385 




Fig. 320. — The 
Daniell Cell. 



heavy bar of zinc Z immersed in a dilute solution of zinc 
sulphate (ZnSO^). The porous cup does not check the 
flow of electricity, but does prevent the rapid mixing of 
the two solutions. Dilute sulphuric acid may be used 
in place of zinc sulphate. 

In this cell the zinc is continually being dissolved and 
in the course of time must be renewed. On the other 
hand, copper (Cu) from the solution of 
copper sulphate (CuSOJ is deposited slowly 
upon the copper plate, which in time grows 
into a massive sheet. In order to maintain 
r« constant supply of copper ions in the 
solution, crystals of copper sulphate are 
added from time to time. Since copper, 
instead of hydrogen, is deposited on the 
copper plate, the nature of the "^late is 
not changed, and, consequently, no polarization takes 
place. On account of the complete non-'polarization of the 
Daniell cell, it is often used when currents of great con- 
stancy are required. 

405. The Gravity Cell. — A dilute solution of zinc 
sulphate has less density than a saturated solution of 
copper sulphate; hence, the two will be 
kept separate by gravity and the former 
will float upon the latter. This fact is 
employed in the so-called gravity cell. Fig. 
321, in which a copper plate lies upon the 
bottom of a jar surrounded by crystals of 
copper sulphate and a saturated solution 
of the same substance. Above this solu- 
tion is one of dilute zinc sulphate which surrounds a 
massive zinc plate. 

If a gravity cell be allowed to stand on an open circuit, 
the liquids slowly mix. In order to prevent this and thus 
26 




Fig. 321. — The 
Gravity Cell. 



386 



A HIGH SCHOOL COURSE IN PHYSICS 



keep the copper sulphate from reaching the zinc j^late, the 
cell must be kept in more or less active operation. While 
producing a current the copper ions are continually mov- 
ing away from the zinc, in which respect the action is the 
same as that of the Daniell cell, of which this cell is 
a modification. Gravity cells are extensively used in 
telegraphy and in circuits where constant currents are 
desired. 

406. The Leclanche Cell. — The positively charged plate 
of the Leclanche (pronounced Le clan'shd') cell, Fig. 322, 
is a bar of carbon O whicli is packed in 
a porous cup together with small pieces 
of carbon and manganese dioxide. The 
porous cup is placed in a solution of 
ammonium chloride (salammoniac) in 
which*stands a bar of zinc Z to serve as 
the negative plate of the cell. 

When the circuit containing a Leclanche 
cell is closed, hydrogen is liberated at the 
carbon ; but, on account of the presence 
of the manganese dioxide, the hydrogen is slowly oxidized, 
forming water. In this manner polarization is largely 
prevented. As a rule, however, the hydrogen is liberated 
so rapidly that the cell slowly polarizes, but regains its 
normal condition when allowed to stand for a time on 
an open circuit. 

The Leclanche cell has had a very extensive use on 
account of the fact that it produces currents that are 
suitable for ringing bells, operating signals, regulating 
dampers, etc. The cell will remain in good condition for 
years with very little attention. At the present time it 
is being rapidly replaced by the more convenient and in- 
expensive " dry " cell, which is a modified form of the 
Leclanche. 




Fig. 322.— The Le 
clauche' Cell. 



VOLTAIC ELECTRICITY 



387 




407. The "Dry " CelL — The Leclanche cell is made in 
the form of the so-called " dry " cell by embedding the car- 
bon plate (7, Fig. 323, in a paste A made by mixing zinc 
oxide, ammonium chloride, 
plaster of Paris, zinc chlo- 
ride, and water. The whole 
mass is contained in a zinc 
cup Z, which serves as the 
negative plate of the cell. 
Evaporation is prevented by 
hermetically sealing the cup 
with melted bitumen or 
asphalt. Many different 
forms of dry cells are now 
on the market and are in 
great demand for operating 
the sparking devices of gas 
and gasoline engines, ring- 
ing bells, etc. A " dry " cell deteriorates rapidly on a 
closed circuit, and hence should always be connected with 
a spring key that automatically opens the circuit when 
the cell is not in use. 

EXERCISES 

1. Explain how the direction of the current in a telegraph wire 
could be determined by means of a small compass. 

2. Is the difference of potential between the plates of a voltaic cell 
large enough to cause a spark when wires from them are brought 
near together? Is it great enough to produce a shock when the plates 
are simultaneously touched? 

3. Would you expect to get an E. M. F. by forming a cell of two 
copper plates or two zinc plates in dilute sulphuric acid ? Make the 
experiment, using a sensitive galvanometer. Try the experiment with 
a polarized copper plate and one that is not polarized and account for 
the results. 

4. Would you expect to derive a current from a zinc and copper 
cell containing a solution of common salt ? Perform the experiment. 



Fig. 323. — The "Dry " Cell. 



388 



A HIGH SCHOOL COURSE IN PHYSICS 



5. Why is the combination of zinc, copper, and dihite sulphuric 
acid a suitable one for experiments like those described? 

6. Study the description of the Daniell cell and state which solu- 
tion grows weaker and must ultimately be renewed. What materials, 
therefore, must be kept on hand for replenishing a system of Daniell 
or of gravity cells ? 

2. EFFECTS OF ELECTRIC CURRENTS 

408. The Magnetic Effect. —It has already been shown 
that a current of electricity is capable of magnetizing bars 
of iron and steel (§ 390), and also that a magnetic needle J 
placed near a current is deflected from its normal position. 
These are only special cases of the more general one illus- 
trated by the following experiments: 

1. Join several new dry cells in series and connect them through 
a key to a vertical wire passing through a horizontal sheet of card- 








'-'u^ 



Fig. 324. — Magnetic Field 
around a Conductor Carrying 
a Current. 




Fig. 325. — Iron Filings Arranged 
around a Conductor. 



board as in Fig. 324. A current of at least 2 amperes is desirable. 
Sprinkle iron filings on the cardboard and close the key for a short 
time, meanwhile tapping lightly to jar the filings. The filings will 
arrange themselves in circular lines (see Fig. 325) around the wire 
as the center, thus showing the shape of the magnetic field about the 
conductor. Small pivoted magnets placed near the wire will turn 




VOLTAIC ELECTRICITY 389 

until they are tangent to the circles with their N-poles as shown in 
Fig. 324. 

2, Make a helix, or spiral, of insulated copper wire by winding 
about 30 turns upon a lead pencil. Connect the ends of the wire with 
a cell and present one end of the helix to the N-pole of a suspended 
magnet or compass needle. One end will be found to attract the 
N-pole while the opposite end repels it, and the end that attracts 
the N-pole will repel the S-pole. Reverse the connections of the cell 
and repeat the experiment. The end of the helix which formerly 
attracted a magnetic pole will 
now repel it. 

3. Thread a spiral of cop- 
per wire through holes in a 
flat piece of cardboard or wood 
as shown in Fig. 326. Strew 
iron filings evenly over the Fig. 326. — Magnetic Field Produced by a 

£ J J ,x^ . Current in a Helix, or Solenoid, 

surface and send the current 

(about 3 amperes are required) from several new dry cells through 

the wire. If the apparatus be tapped lightly, the filings will arrange 

themselves as shown in the figure. 

It is clear from these experiments that a conductor 
carrying an electric current is surrounded by a magnetic 
field. If the conductor is a straight wire, Experiment 1 
shows that the lines of force are concentric circles around 
the conductor as the center. The direction of the lines is 
given by the direction in which an N-pole is urged when 
placed in the field. It is clear, therefore, from Fig. 324, 
that the direction of the lines is that indicated by the 
arrows. A convenient rule may be stated as follows : 

Grasp the conductor with the right hand with the out- 
stretched thumb in the direction the current is flowing. The 
fingers encircle the ivire in the direction of the lines of force. 

The shape of the field depends on the form of the con- 
ductor ; for, when the conductor is a helix, the magnetic 
field resembles tliat about a straight bar magnet. (See 
Fig, 327.) In fact, it is shown in Experiment 2 that the 



390 



A HIGH SCHOOL COURSE IN PHYSICS 



helix has the properties of a magnet while a current is 
flowing through the wire. If the coil could be properly 
suspended and a current sent through it, the axis would 
assume the direction taken by a compass needle. Such a 
helix is also called a solenoid. 




Fig. 327. — Showing the Relation between the Direction of the Current and 

the Magnetic Lines. 

409. Poles of a Helix. — Let Experiment 2 of § 408 be repeated 
and the N-pole and the S-pole ascertamed. Tracing the current from 
the positive pole of the cell, it will be found to flow around the N-pole 
in a direction contrary to the motion of the hands of a clock as one 
faces the pole and in the reverse direction about the S-pole as shown 
in Fig. 327. 

The experimental result leads to the following conven- 
ient rule : If the helix he grasped ivith the right hand so that 

the fingers point in the direction 
the current is floiving^ the ex- 
tended thumb will poiyit in the 
direction of the N-pole of the 
helix. (See Fig. 328.) 

Another convenient rule is 
applied by facing the end of the 
helix ; if the current is flow- 
ing clockwise, the end of the helix is an S-pole, and, 
conversely, if counter-clockwise, an N-pole. 




Fig. 328. — Rule for Determining 
the Poles of a Helix Carrying 
a Current. 



VOLTAIC ELECTRICITY 391 

410. The Electro-magnet. — Probably none of the effects 
that can be produced by electric currents are of greater 
practical value or employed more extensively than the 
magnetic effect. The scale on which this effect can be 
produced is limited only by the dimensions of the appa- 
ratus used and the strength of the current. 

Wind a hehx consisting of about 75 turns of No. 22 insulated 
copper wire and provide an iron core that can be inserted into, or 

removed from, the helix as de- ^,^^1^ 

sired. Test the magnetic action ,.<«^^^^^^V^-<i.-i---^ 

of the helix without the core ,^^81^^^ V '^'^ 

by presenting it to a magnetic ^^^0^ ^s^«-*-« — ^^^ 

needle while a current is flow- c^^L I 

ing. Now insert the core and ■^w I i 

note the change. Its effect is ^ |j| j j^^i. 

more marked than before. Next — u^^ 

send the current from a new ^^°- 329. - Magnetization of Iron by 

, n ,, 1 ,1 1 T 1 Means of an Electric Current, 
dry cell through the helix and 

dip one end of the core into a box of tacks, Fig. 329. A large 
quantity of tacks will cling to the core and remain there until the 
current is interrupted. 

Although a coreless helix of wire has magnetic proper- 
ties when a current is flowing through it, its magnetic 
field is insignificant when compared to that which the 
same current will produce when an iron core is present, 
on account of the large permeability of iron (§ 381). 
The introduction of the core adds the lines of force of 
the magnetized iron to those produced by the current in 
the helix alone. Any mass of iron around which is a helix 
for conducting an electric current is called an electro-magnet. 
The small amount of magnetism retained by the core after 
the circuit is broken is termed residual magnetism. 

Electro-magnets are made in many forms and often of 
extremely large dimensions for holding heavy masses of 
iron. The horseshoe form shown in Figs. 330 and 331 is 



392 



A HIGH SCHOOL COURSE IN PHYSICS 



most frequently used in electrical devices. The wire is 
so wound as to produce an N-pole at iV and an S-pole 
at S. The bar of iron A which is held by the magnetism 
of the poles is called the armature. When 
the armature is against the poles, it will 
be observed that the lines of force find a 




L !'j{''':-r=^ 



ilUII. 




Fig. 330.— An Electro-magnet 
Showing Poles and Armature. 



Fig. 331. — Showing 
the Path of the 
Lines of Force in 
an Electro-magnet. 



Fig. 332. — A Large 
Electro-m a g n e t 
Used for Han- 
dling Masses of 
Iron in Factories. 



complete magnetic circuit through iron, as shown by 
the dotted lines in Fig. 331. Figure 332 shows a large 
form of the electro-magnet that is widely used in manu- 
facturing plants for the purpose of handling heavy masses 

of iron, as castings, 
plates, pig iron, etc. 
The lifting power is con- 
trolled mainly by the 
current used. 

411. The Electric Bell. 

— An important application 
of the magnetic effect of an 
electric cm-rent is found in 
the electric bell. The instru- 
ment consists of an electro- 
magnet £", Fig. 333, near the 
poles of which is an armature 
of iron attached to a spring. 
Extending from the armature 
Fig. 333. — The Electric Bell. is a slender rod bearing at its 




VOLTAIC ELECTRICITY 



393 



extremity the bell hammer H. The armature carries a spring that 
touches lightly against the screw point at C. The connections are made 
as shown in the figure. When the push button P is pressed, the circuit 
is completed by a metallic contact within the button, and the current 
from the cell B flows through the electro-magnet coils. This causes 
the magnet to attract the armature, and the hammer strikes the bell. 
The movement of the armature, however, breaks the circuit at C and 
thus interrupts the current. Since the cores of the magnet now lose 
their magnetism, the armature is thrown back by the spring ; the 
contact at C is restored, and all the operations are repeated. Hence 
a steady pressure on the push button causes the hammer to execute a 
number of rapid strokes against the bell. The direction of the cur- 
rent is immaterial to the operation of the bell. 

412. Mutual Action of Two Parallel Currents. — Suspend 

two light wires about 30 cm. long from two other wires a and h, 

Fig. 334, which are bent as 

shown. Let the lower ends of 

the vertical wires just dip into 

a mercury cup below. Now send 

a strong current through the ap- 



5 




!l 




feTira^ 




Fig. 334. — Illustrating the Mutual 
Action between Parallel Currents. 



Fig. 335. — Parallel Currents in the 
Same Direction. 



paratus (an alternating current of from 4 to 8 amperes suffices), which 
will flow down one wire and up the other. The two conductors will 
repel each other. Again, hang both wires on a and make the con- 
nections as shown in Fig. 335. Both currents will now flow in the 
same direction and an attraction will be observed. 



394 



A HIGH SCHOOL COURSE IN PHYSICS 



The results of the experiments may be stated as follows: 
Currents flowing in the same direction attract each other^ 
and those flowing in opposite directions repel each other. The 





(1) 

Fig. 336. — Magnetic Fields around Parallel Conductors: (1) When Currents 
are Flowing in the Opposite Directions ; (2) When Flowing in the Same 
Direction. 



magnetic field around the two wires in each of the two 
cases is shown in Fig. 336. 

413. Heating Effect of Electric Currents. — It is a familiar 
fact that electric currents produce heat. This is at once 
evident when the hand is placed in contact with the warm 
bulb of an incandescent lamp. The following experiment 
shows in another way the transformation of electrical 
energy into heat. 

If an electric lighting current is available, construct a device for 
controlling it by mounting several lamp sockets on a board and join- 
ing them in parallel (§444). Connect each side to a binding post, 
screw a lamp into each socket, and a safe adjustable resistance is 
ready for use. Put the device just described in circuit with a piece 
of fine iron wire joined to one of copper wire of the same diameter. 
The pieces may be each several inches in length. Screw one lamp at 
a time into its socket until sufficient current flows to heat the wire. 
The iron wire will at length glow with heat while the copper wire of 
the same dimension is still comparatively cool. If a lighting current 
cannot be used, the experiment may be made by using shorter pieces 
of wire with several good dry cells joined in series. In this case no 
controlling resistance is necessary. 

An experiment that succeeds well with a small current is made by 
connecting a cell with a fine insulated copper or iron wire that has 



VOLTAIC ELECTRICITY 



395 



been wound in a coil about the bulb of a thermometer. 
mercury indicates the heating effect of the current. 



The rise of 



The heating effect of a current of electricity is employed 
in electric cooking devices, in various methods of heating, 
and in electric lighting. 
An electrically heated 
fiatiron is shown in Fig. 
337. 




Fig. 337. — An Electrically Heated Flatirou. 



414. Chemical Effects 
of Electric Currents. — 

Seal two platinum wires to 
which are attached platinum 
strips about 1 cm. wide and 

3 cm. long in small bent glass tubes e and /, Fig. 338. Over each 
of these place an inverted test tube filled with water to which has 
been added a number of drops of sulphuric acid. Place a small 

quantity of mercury in each of the small 
tubes and connect the strips of platinum 
in circuit with three or four dry cells 
joined in series by inserting the con- 
necting wires into the mercury. Small 
bubbles of gas will be seen rising from 
the platinum in each tube and collect- 
ing at the top. If e is joined to the 
positive pole of the battery, the gas 
above that strip of platinum will collect 
only one half as fast as that over the 
other. When the action has progressed until one of the tubes 
is filled with gas, remove it carefully and apply a lighted match. 
A blue flame will be seen caused by the burning of this gas, which is 
hydrogen. Now remove the other tube and insert a glowing pine 
stick, which will be observed to burst into a flame because of the 
oxygen contained in the tube. 

The platinum strips employed in making electrical con- 
tact with the liquid in the tubes are called electrodes. 
While conducting the electric current from one electrode 
to the other, the water is decomposed into its constituent 




Fig. 338. — Electrolysis of 
Water. Oxygen is Collected 
at a, and Hydrogen at 6. 



396 A HIGH SCHOOL COURSE IN PHYSICS 

elements, hydrogen and oxygen. This decomposition is 
characteristic of all liquids, except liquid metals. The 
process of decomposing a compound substance hy means of 
an electric current is called electrolysis. The substance 
decomposed is called an electrolyte. The electrode at 
which the current enters the electrolyte is the anode; 
the one at which it leaves, the cathode. 

415. Theory of Electrolysis. — In the hght of the theory of 

solutions stated in § 396, the process of electrolysis is easily explained. 
When a small quantity of sulphuric acid (HgSOJ, for example, is 
introduced into water, the molecules split up into hydrogen (H) ions 
and sulphions (SO^), the former bearing positive charges of electric- 
ity, the latter, negative charges. ISTow when two platinum electrodes 
that are connected to the poles of a battery are placed in the liquid, 
the cathode, which is charged negatively, attracts the positively 
charged H ions, while the anode, which is positively charged, attracts 
the negatively charged SO^ ions. The H ions discharge their positive 
electricity to the cathode and are then free to collect and rise in 
bubbles to the surface. The SO^ ions, on the other hand, discharge 
their negative electricity to the anode, where they react chemically 
upon the water (HgO), setting free the oxygen (O). Thus each 
sulphion (SO4) together with the hydrogen (Hg) taken from the 
water forms new molecules of sulphuric acid (H.^SO^). In this manner 
the amount of acid present in the solution remains constant, while the 
quantity of water diminishes as the process of electrolysis advances. 

416. Electrolysis of Copper Sulphate. — Introduce two plati- 
num electrodes into a solution of copper sulphate (CuSO^) and place 
them in circuit with three or four dry cells. After a few seconds the 
cathode, or negative electrode, will be found to be coated with me- 
tallic copper, while the anode remains unchanged. If the direction of 
the current be now reversed, copper will be deposited on the clean 
plate (now the cathode), while the copper coating on the anode gradu- 
ally disappears. 

In the electrolysis of copper sulphate, the ions present are copper 
(Cu) ions and sulphions (SO^), the former bearing positive charges 
of electricity and the latter, negative. When platinum electrodes are 
introduced into the solution, the copper ions are attracted to the 
cathode, where they discharge their electricity and become free. Thus 



VOLTAIC ELECTRICITY 



397 



we find copper deposited upon this electrode. At the anode the sul- 
phions react with water as in the case just described (§ 415). How- 
ever, if the anode is a copper plate, the sulphions (SO^) abstract copper 
from the plate at the instant they discharge their electricity and form 
copper sulphate (CuSO^), which dissolves in the liquid. 

The experiment illustrates the process of electroplating, 
a method by which one metal is given a coating of another. 
Thus by electrolytic action corrosive iron may be plated 
with non-corrosive nickel, or tarnishing brass with pure 
gold. 

In the experiment, copper (Cu), which is a constituent 
of the copper sulphate (CuSO^) in the solution, is always 
deposited on the cathode, or negative electrode. If the 
anode is made of platinum, oxygen will be liberated at its 
surface. If, however, the anode is made of copper, the 
case is modified ; for, instead of setting oxygen free, 
copper is removed from the plate and carried into the 
solution. 

417. Electroplating. — When we consider the vast num- 
ber of plated articles in everyday use, we can scarcely 
overestimate the great commercial value of the electrolytic 
action of an electric cur- 
rent. When, for ex- '^^^'^ ™==-^^^*. + 
ample, silver is to be 
plated upon the surface 
of a spoon, an anode 
plate of silver is sus- 
pended in a solution of 
silver cyanide, while 
the spoon is made the 

cathode by being connected with the zinc of a battery 
and completely submerged in the solution. See Fig. 
339. The current is allowed to flow until the coating 
is of the desired thickness. In the nickel-plating pro- 




FiG. 339. — Illustrating the Process of Elec- 
troplating. 



398 A HIGH SCHOOL COURSE IN PHYSICS 

cess an anode of nickel is used in a solution of nickel 
nitrate and ammonium nitrate. The article to he plated 
is always the cathode. When the coating reaches the 
proper thickness, the final process of polishing gives the 
surface the desired appearance. 

418. Electrotyping. — As a rule, books of which a large 
edition is to be printed are first electrotyped. In this 
process an impression is made in wax after the type has 
been set up, so that each letter leaves its imprint in the 
mold. A thin layer of finely powdered plumbago, or 
graphite, is brushed over the surface of the wax in order 
to render it a conductor of electricity. When thus pre- 
pared, the mold is placed in an electrolytic bath of copper 
sulphate and joined to the negative pole of a battery or 
other source of electricity. The anode is simply a copper 
plate. The current is allowed to flow until the coating 
of copper upon the wax is somewhat thicker than a sheet 
of paper. While the copper is being deposited upon the 
conducting graphite, it penetrates into even the smallest 
depressions of the mold and thus reproduces in copper 
the exact form of the type. The coating of copper is 
then removed from the wax, trimmed, and filled in at 
the back with molten type metal. The advantage gained 
by electrotyping is convenience, durability, and perma- 
nence, and the type from which the impression is taken 
on the wax may be distributed and used again without 
delay. 

419. Refining Copper. — Copper as it comes from ordi- 
nary smelting works contains many impurities. Such 
copper is refined electrolytically by casting the crude 
metal in huge plates which are afterwards used as anodes 
in large depositing vats. The solution used is copper 
sulphate, and the cathode is a thin plate of pure copper. 
When a current of electricity is sent through the solution, 



VOLTAIC ELECTRICITY 



399 



copper is deposited on the cathode until it grows into a 
heavy plate. The copper anode is carried into the solu- 
tion, while its impurities collect at the bottom of the vat. 
Copper thus refined is called electrolytic copper^ and is 
much used in the manufacture of wire and in the construc- 
tion of dynamos, motors, etc. 

420. The Storage Battery. — The principle of the storage cell 
may be illustrated by making a small cell of two plates of lead about 
2x6 inches and a solution of sulphuric acid consisting of one part of 
acid and about eight parts of water. Attach the plates to a piece of 
wood and hang them in the solution. Connect the lead plates to two 
good dry cells joined in series and allow the current to flow for a minute 
or more. Disconnect the dry cells and run wires from the lead plates 
to an electric bell. The bell will ring vigorously for a short time and 
then gradually cease. The power of the cell can be restored by again 
connecting it with the dry cells. If a galvanometer be introduced in 
the circuit, it will be found that the discharging current flows in 
opposition to the current used in charging. If the E. M. F. of the 
charged plates is measured by a voltmeter, it will be found to be 
about two volts. 

The charging current decomposes the water as in § 414. Hydrogen 
is liberated at the cathode plate ; but the oxygen produced at the 
anode changes the surface of the plate /ro?n lead (Ph) into lead peroxide 
(^PhOi), which may be recognized by its brownish color. AVhen the 
plates are connected with the electric 
bell, a current flows from the peroxide 
plate through the bell to the other, 
which is simply lead. The current 
continues until the thin coating is 
used up. 

It should be observed that the 
storage cell stores energy but not elec- 
tricity. The work done by the charg- 
ing current results in the production 
of the energy of chemical separation 
in the cell. When the circuit is 
closed, this amount of potential energy 
is transferred to the bell, where it ap- yig. 340. — A Storage Cell, or 
pears as mechanical energy which is Lead Accumulator. 




400 A HIGH SCHOOL COURSE IN PHYSICS 

finally dissipated as sound and heat. A complete storage cell is 
shown in Fig. 340. 

EXERCISES 

1. What would be the effect produced upon the strength of the 
poles of a bar magnet if it were placed in a helix in which the direc- 
tion of the current around the N-pole of the magnet was counter- 
clockwise ? In which it was clockwise ? 

2. A helix is suspended so as to turn freely in a horizontal plane 
and is placed above a strong bar magnet. If a current be sent through 
the helix, what will be its direction around that end which stops over 
the S-pole of the magnet ? 

3. Place a compass box upon one of the rails of an electric railway 
running north and south and see if you can detect the presence and 
direction of a current. 

4. Would you expect a compass needle to point north and south 
in a moving trolley car? Why? 

5. Which is most readily magnetized v^hen placed in a helix, iron 
or steel? Which will retain the greater amount of magnetism? 
How could you produce a permanent magnet by the help of a dry 
cell and a helix? 

6. How could the direction of an electric current be determined by 
means of an electrolytic cell through which it could be caused to flow ? 

7. What change would finally occur in the copper sulphate solu- 
tion in an electrolytic cell having platinum electrodes if the current 
were allowed to flow? 

8. Would the result in Exer. 7 be at all modified if the anode 
were copper? Explain. 

9. Electric circuits in buildings are protected against too strong 
currents by lead wire fuses of the proper size placed in the circuit. 
If by accident the current becomes strong enough to be unsafe, the 
wire melts. Explain in full. If possible, inspect the wiring of some 
building and report on the form in which the fuses are made. 

SUMMARY 

1. An electric current is a continuous discharge, or 
movement, of electricity (§ 390). 

2. Electric currents may be produced by chemical 
action, as in voltaic cells. It may be shown tliat one of 
the terminals of such a cell is charged with positive elec- 
tricity, the other with negative (§§ 391 and 392). 



A 



VOLTAIC ELECTRICITY 401 

3. The E. M. F. of a cell is the difference of potential 
between these charges when no current is flowing (§ 393). 

4. Much of the potential energy of the zinc of a cell 
is wasted by "local action." This may be largely pre- 
vented by amalgamation (§ 397). 

5. A current flowing in a conductor near and parallel 
to a magnetic needle tends to deflect it from its position. 
This principle is used in many electrical measuring instru- 
ments, as galvanometers, etc. (§ 400). 

6. The E. M. F. is diminished by the accumulation of 
hydrogen on the copper (or carbon) plate. This effect 
is known as polarization. (§ 402). 

7. An electric current is surrounded by a magnetic 
field. When the conductor carrying the current encircles 
a bar of iron or steel, the bar becomes a magnet (§ 408). 

8. The electro-magnet consists of a helix of insulated 
wire wound upon a core of iron. The principle of the 
electro-magnet is employed in the electric bell and many 
other important devices (§§ 410 and 411). 

9. Parallel currents flowing in the same direction 
attract each other, and those flowing in opposite directions 
repel (§ 412). 

10. When an electric current flows through a con- 
ductor, the conductor is heated. This effect is applied in 
electric lighting, in heating and cooking devices, etc. 
(§ 413). 

11. An electric current decomposes water into hydro- 
gen and oxygen, hydrogen being liberated at the cathode. 
When a current flows through a solution of a metallic 
salt, the compound is decomposed and the metal liberated 
at (or plated upon) the cathode. This effect of an electric 
current is known as electrolysis and is used in electro- 
plating, etc. (§§ 414 to 420). 

27 



CHAPTER XIX 
ELECTRICAL MEASUREMENTS 

1. ELECTRICAL QUANTITIES AND UNITS 

421. Fundamental Electrical Magnitudes. — In every 
electric circuit there are three fundamental quantities 
which admit of measurement ; viz. current strength^ elec- 
trical resistance^ and difference of potential. The first of 
these, current strength, may be likened to the rate at 
which water is delivered through a pipe ; electrical resist- 
ance, to the friction encountered by a liquid current ; 
and difference of potential, to a difference of level (or 
pressure) for any two chosen points between which the 
current is flowing. 

422. Current Strength — the Ampere. — In many of the 
preceding experiments it has been obvious that the mag- 
nitude of many of the effects produced depended on a 
quantity that has been frequently referred to as the 
"strength of the current." For some effects the current 
must be strong, for others, weak. The expression refers 
to the rate at which positive electricity is being discharged 
from the positive to the negative pole through the circuit. 

The unit of current strength is the ampere^ so called in 
honor of Ampere,^ a French physicist. The ampere is that 
current which will deposit in an electrolytic cell 0.001118 
grams of silver or 0.0003287 grams of copper per second. 
Of the currents used in the experiments described in the 
preceding sections, the largest were required in § 413 and 

1 See portrait facing page 406. 
402 



ELECTRICAL MEASUREMENTS 



403 




amounted to 6 or 7 amperes. As a rule, the current used 
in most classroom demonstrations is less than 1 ampere in 
value. In the comparison and measurement of electric 
currents, different kinds of galvanometers are used. 

423. The Tangent and Astatic Galvanometers. — The tan- 
gent galvanometer is a common form found in most laboratories. 
It consists of a circular coil of wire wound on a frame about 30 cm. 
in diameter. See 

(1), Fig. 341. The 

ends of the coil 

lead to binding 

posts on the base. 

At the center of 

the coil is placed 

a compass needle 

below which is a 

graduated circle. 

When in use the 

coil of the instru- 
ment is placed 

north and south and 

the current sent through the coil. The instrument derives its name 

from the fact that the current is proportional to the tangent of the angle 

of deflection. If the current necessary to deflect the needle 45° is 

known, other currents are easily measured. 

For the measurement and detection of small currents the astatic gal- 
vanometer (2), Fig. 341, is sometimes used. 
Two similar magnets are attached to a ver- 
tical rod so that their N-poles point in op- 
posite directions. This system is then 
suspended so that the lower needle swings 
within a coil of wire, as shown in Fig. 342. 
When a current is sent through the coil, all 
parts of it tend to throw the needles out of 

Fig. 342. — The Arrange- the north-and-south positions. This form of 
ment of the Magnets in galvanometer may be made extremely sen- 
an Astatic Galvanometer. ^.^.^^ ^^ ^^^^^^^ currents. 

424. The d'Arsonval Galvanometer. — Galvanometers of the 
d'Arsonval type differ from those described in § 423 in that the 



Fig. 



341. 
(2) 



(2) 



— (i) The Tangent Galvanometer ; 
The Astatic Galvanometer. 




404 



A HIGH SCHOOL COURSE IN PHYSICS 



magnet is stationary and the coil of wire movable. As shown in 
Fig. 343, the instrument consists of a light coil of many turns of fine 

wire, one end of which is the 
supension wire AC, while 
the other end extends below 
the coil and connects at B. 
AVhen the instrument is 
joined in an electric circuit, 
the current is conducted 
through the wire of the coil. 
When no current is flowing, 
the plane of the coil is held 
by the suspending wire par- 
allel to a line joining the 
two poles of a permanent 
magnet N and S. Since a 
current through the coil de- 
velops a magnetic field of its 
own at right angles to that of 
the magnet, the coil will turn 
until the maa:netic forces are 




Fig. 343. — The d' Arson val Galvanometer. 



in equilibrium with the torsional resistance of the suspension wire. 
The deflections are read by the movement of a beam of light reflected 
from a small mirror M attached to the coil. For small angles of de- 
flection the current is practically proportional to the angle ; hence the 
instrument may be used in current measurements. The instrument 
also affords a very sensitive detector of currents and is, on this account, 
indispensable in a large class of experiments. 
The principle involved in this galvanometer 
is employed in many electrical measuring 
instruments. 

425. The Ammeter. — An instrument 
for measuring the strength of an electric cur- 
rent is called an ammeter, or amperemeter. In 
most ammeters the magnetic effect of a cur- 
rent is employed. The instrument may con- 
sist simply of a magnetic needle, which shows 
by the amount of its deflection the number 
of amperes of current. The best instruments, however, consist of a 
delicately pivoted coil of wire A, Fig. 344, turning between the poles 




Fig. 344. — Section of au 
Ammeter. 



ELECTRICAL MEASUREMENTS 



405 




of a strong permanent magnet iV and S and held in position by two 
hair springs a and b. The principle involved in this class of instru- 
ments will be recognized at once as that of the d'Arsonval galvanom- 
eter. When a current is 
sent through the coil, the 
magnetic field developed by 
the current causes the coil 
to turn, and the pointer p 
moves over a scale which is 
graduated to read in am- 
peres. When the current 
is interrupted, the coil is 
restored to its initial posi- 
tion by the springs which 
also serve to conduct the 
current into and out of the Fig. ai5. — Au Ammeter, 

coil. The complete instrument is shown in Fig. 345. 

426. Electrical Resistance. — It was observed in § 350 
that substances differ in respect to the readiness with 
which tlie}^ transmit an electrical charge ; thus bodies are 
classed as good or poor conductors. Tlie opposition that a 
conductor offers tending to retard the transmission of elec- 
tricity is called electrical resistance. Hence, with a given 
source of electricity, as a Daniell cell, the current strength 
will diminish as the resistance of the circuit is increased, 
and will rise in value as the resistance is decreased. 

427. Laws of Resistance. — Construct a frame about 1 meter 
long and upon it stretch 4 wires terminating in binding posts. Let 
No. 1 consist of 1 meter of No. 30 (diameter 0.010 inch) German silver 
wire ; No. 2, of 2 meters of No. 30 German silver wire ; No. 3, of 2 meters 
of No. 28 (0.013 inch) ; and No. 4, about 20 meters of No. 30 copper 
wire. Connect wire No. 1 in series with one or two Daniell cells and 
a low resistance galvanometer, and read the deflection of the needle. 
Replace No. 1 by No. 2, and the deflection will be found to be less 
than before, thus indicating a greater resistance for the greater length. 
When the current is sent through No. 3, which is a lai-ger wire, an 
increased deflection shows a decreased resistance. Finally, when the 
current is sent through wire No. 4, the deflection will be even larger 



406 A HIGH SCHOOL COURSE IN PHYSICS 

than for No. 2, which is of the same size and only one tenth as long. 
Thus copper is shown to be more than 10 times as good a conductor 
as German silver. The experiment may be extended to other wires 
of various substances. 

Accurate measurements verify the following laws : 

1. The resistance of a conductor of uniform size and com- 
position is directly proportional to its length. 

2. The resistance of a conductor is inversely proportional 
to its cross- sectional area; or^ if circular in form^ to the 
square of its diameter. 

3. The resistance of a conductor depends upon the nature 
of the substance of which it is composed. 

For example, if the resistance of a copper wire of a 
certain diameter and length is 1 unit, the resistance of a 
wire of the same kind and size and twice the length is 
2 units. If, now, the diameter be doubled, the resistance 
is divided by 2^, i.e. reduced to 2 units -j- 4, or | a unit. 

428. The Unit of Resistance. — The unit of resistance 
is called the ohm in honor of Dr. G. S. Ohm,^ a German 
physicist. The ohm is the amount of electrical resistance 
offered by a column of pure mercury 106.3 cm. in height^ 
of uniform cross section., and having a mass of 14.4521 
grams., the temperature being 0° O. The cross-sectional area 
of such a column is almost exactly one square millimeter. 
Since such a column of mercury would be inconvenient to 
handle, coils of wire whose resistances have been carefully 
measured and recorded are used in practical measure- 
ments. A piece of No. 22 (0.025 inch) copper wire 60 
feet in length lias approximately one ohm of resistance. 
One meter of No. 30 German silver wire has a resistance 
of about 6.35 ohms. A convenient unit for rough work 
can easily be made by winding 9 feet and 5 inches of 

1 See portrait facing page 406. 



ANDRE MARIE AMPERE (17T5-1836) 



The fame of Ampere rests mainly on 
the services he rendered to science in es- 
tablishing the relation between electricity 
and magnetism and in developing the sci- 
ence of electro-dynamics. His chief experi- 
ments deal with the magnetic action be- 
tween conductors in which electric currents 
are flowing. Ampere was also the first to 
niaanetize needles bv inserting them in a 
helix in which a current was flowing. 

Ampere was born at Lyons, France. 
During the French Revolution his father 
was beheaded, an event which for years 
clouded the spirit of the young scientist. 
In 1805 he became professor of mathe- 
matics in the Polvtechnic School in Paris, and later professor of 
physics in the College of France. In 18^3 he published his mathe- 
matical classic on the theory of magnetism. The practical unit of 
current strength is named in his honor. 




GEORGE SIMON OHM (1T8T-1854) 



Following closely upon the experiments 
of Ampere on the magnetic action between 
currents came the researches of Ohm, a 
German, whose discoveries concerned the 
strength of an electric current. 

Ohm's first experiments dealt wdth the 

conductivity of wires Of diflPerent metals. 

He observed the deflections of a magnetic 

needle by currents from a given source, 

but flowing through difl'erent conductors. 

By changes in the E. M. F. in the circuits 

used, he obtained results which led him 

E 
to the well-known expression C = ^ . 

He then investigated cells joined in paral- 
lel and series, and published his results in 1836. During the following 
year he published the theoretic deduction of the law bearing his name. 
This law is one of the most fruitful of the early electrical discoveries. 
Ohm was born in Erlangen, where he was educated. His ambi- 
tion to become a university professor was not realized until 1849, 
when he w^as elected to a professorship at Munich. The practical 
unit of resistance is called the ohiyi in his honor. 




ELECTRICAL MEASUREMENTS 407 

No. 30 copper wire on a spool and connecting its ends to 
binding posts. 

429. Potential Difference — the Volt. — It was shown 
in § 393 that the difference of potential between the poles 
of a voltaic cell when no current is flowing is its elec- 
tromotive force. This quantity is conceived to be the 
cause of current flow, and is analogous to the difference 
in level between two bodies of water, to the difference in 
pressure of gases compressed in two reservoirs, or to a 
difference of temperature in the case of heat. The unit 
of potential difference and, consequently, of E. M. F., is 
the volt. The volt is that difference of potential between the 
ends of a wire having a resistance of one ohm which will 
'produce a current of one ampere. The E. M. F. of some 
of the voltaic cells in common use is given in the follow- 
ing table : 

E. M. F. OF Cells 

Daniell, 1.1 volts. Dry cells, 1.5 volts. 

Leclanche, 1.5 volts. Dichromate, 2.0 volts. 

A Daniell cell would then cause a current of 1.1 am- 
peres through a wire having a resistance of 1 ohm, pro- 
vided there were no other resistance in the circuit. It is 
evident, however, that the resistance of the cell itself 
must always be considered in any given circuit. 

430. The Voltmeter. — lii order that an instrument may 
measure the E. M. F. of a cell, it is necessary that no ap- 
preciable current be permitted to flow. See the definition 
of E. M. F., § 393. In order, therefore, that a galvanom- 
eter may be adapted to this use, it must contain a large 
amount of resistance. Instruments of several hundred 
ohms are made whose deflections are practically propor- 
tional to the E. M. F. of cells to which they may be 
attached. When such instruments are graduated to read 
volts they are. called voltmeters. Figure 346 shows the 



408 



A HIGH SCHOOL COURSE IN PHYSICS 



manner of connecting a voltmeter for determining the 
E. M. F. of the cell B, and Fig. 347 shows its connection 




Voltmeter 




Fig. 346. — Illustrating Fig. 347. — The Voltmeter Shows the 

a Use of the Volt- Fall of Potential through Coil C and 

meter. the Ammeter Measures the Current 

Strength. 

for giving the potential difference between the terminals 
of a coil of wire placed in the circuit. One well-known 

form of a voltmeter is 
shown in Fig. 348. 

431. Ohm's Law. — Connect 

a Daniell cell in circuit with a 
galvanometer or ammeter and a 
resistance box. Make the resist- 
ance of the circuit a suitable 
amount (say 200 ohms, including 
the galvanometer resistance) and 
measure the current strength. 
Replace the Daniell cell by a new 
dry cell and again ascertain the current. The two currents will be 
found proportional to the E. M. F. of the cells as stated in the table in 
§ 429. Again introduce the Daniell cell and make the resistance of 
the entire circuit double the first amount. The current strength will 
be only one half as great. Make the resistance one half as great as at 
first, and the current will be doubled. 

The experiment illustrates the law first published by 
Dr. G. S. Ohm in 1827. This law states that the strength 
of an electric current is directly proportional to the U. Jf. F* 




Fig. 348. — The Voltmeter, 



ELECTRICAL MEASUREMENTS 409 

furnished hy the voltaic cell or combination of cells, and in- 
versely proportional to the total resistance of the circuit. 
The ampere, volt, and ohm are so chosen that Ohm's law 

may be written 

E 

i.e. current (amperes) = —^ — ^ , — ^ — (i) 

resistance (ohms) 

The law may be applied to any portion of an electric 
circuit, in which case it becomes 

. ^ X potential difference (volts) .^^ 
current (amperes) = ^^ (2) 

resistance (ohms) 

For example, the current produced by a Daniell cell 
(1.10 volts) in a circuit in which the total resistance 
(including that of the cell) is 41 ohms is l.l-j-44, or 
0.025 ampere. It is clear from equation (l) that if any 
two of the quantities are given, the third can easily be 
computed. 

432. Internal Resistance. — The current that any cell 
can produce is limited by the resistance which the current 
encounters in passing through the liquid of the cell. This 
is called the internal resistance of the circuit. In a fresh dry 
cell the resistance is a fraction of an ohm, but with age and 
use it increases to several ohms. That of a Daniell cell varies 
from about one to three or four ohms. The resistance of cells 
may be decreased by using larger plates and also by reducing 
the distance between the plates. It is furthermore depend- 
ent on the conductivity of the liquid. When more than 
one cell is connected in a circuit, the entire internal resist- 
ance depends on the manner in which they are joined 
together, 




410 A HIGH SCHOOL COURSE IN PHYSICS 

433. Cells Connected in Series. — Join each of three Daniell 
cells successively to a voltmeter, or a galvanometer of high resistance 
(not less than 50 ohms), and record the deflections produced. They 

should be about equal. Now connect two 
of the cells in series, as shown in Fig. 349, 
and lead wires to the instrument. The de- 
flection will be twice as great as for one cell, 
and the addition of the third cell will make 
the current three times as strong. The de- 
flections under these conditions are propor- 
FiG. 349. — Two Cells tional to the number of cells, because their 
Joined in Series. electromotive forces are added together when 

they are joined in this manner. 

Oells are in series when the positive pole of the first is 
joined to the negative pole of the second^ the positive pole of 
the second to the negative pole of the thirds and so on. A 
study of the figure will aid greatly in making the method 
clear. See also Fig. 311. 

Although the electi'bmotive forces are added by con- 
necting cells in series, the resistances of the separate cells 
also add together and thus tend to reduce the current 
strength. For example, four similar cells in series will 
have four times the internal resistance of one cell. 

Since the strength of the current is obtained by divid- 
ing the E. M. F. by the total resistance of a circuit, 

^^ E^+E. + E.-f E^-f etc. ^ ^3. 

R + r^ + r2 + r3 -f- r^ -f etc. ' 

where E^^ E^, E^^ E^^ etc. are the electromotive forces of 
the individual cells, r^, r^^ ^31^41 etc. are the correspond- 
ing internal resistances and R is the external resistance of 
the circuit. If we have, for example, five similar cells, 
the equation becomes 

C = ^^ , and for n cells, C = ^^ , (4) 

R + 5r R + nr ^^ 



ELECTRICAL MEASUREMENTS 



411 




where E is the E. M. F. and r the resistance of a single 
celL 

434. Cells Connected in Parallel. — i. With the apparatus 
used in the experiment of the preceding section, record the deflec- 
tion produced by each 
cell alone and then con- 
nect the positive poles of 
two of them to one termi- 
nal of the instrument and 
the two negative poles to 
the other terminal. The 
deflection will be the same 
as that obtained when 
only one cell is used. If 
all the cells are joined as 
shown in Fig. 350, the de- Fig. 350. — Four Cells Joined in Parallel, 
flection of the instrument will not be greatly increased, if at all. 

2. Join the cells while connected in parallel to an ammeter or 
galvanometer of very small resistance and record the reading. Do 
the same with the cells connected in series and also with only one cell. 
It will be found that one cell alone will produce about as much cur- 
rent as all the cells when joined in series, but a much stronger current 
will be derived from them when they are in parallel. 

The experiments show that the parallel arrangement of 
cells has an advantage over the series connection when the 
external resistance is small. In fact, when the external 
resistance is very small, as it is in some cases, the series 
arrangement of cells produces practically no more current 
than a single cell.^ The following example will make the 
matter clear. 

The current obtained from a single cell whose E. M. F. is 1.5 volts 
when the internal resistance is 3 ohms and the external resistance 

1.5 



0.2 ohm is, by equation (1), C = 



or 0.469 ampere. 



0.2 + 3 

1 When cells of extremely small internal resistance are used, as new 
dry cells, the series arrangement is the better under nearly all circum- 
stances. A poor cell, however, having a large resistance may prove to be 
more of a hindrance than a help. 



412 A HIGH SCHOOL COURSE IN PHYSICS 

Now if ten such cells are connected in series, the current is, by 

10 X 1 5 
equation (4), C = ^, or 0.496 ampere, which is only slightly 

larger than the current obtained from one cell. 

Experiment 1 shows that the E. M. F. remains un- 
changed when cells are connected in parallel ; hence, 
for the ten cells just considered, the E. M. F. is just the 
same as that of one cell^ viz. 1.5 volts. Again, since like 
plates are joined together, the entire combination of cells 
is like one cell having plates of ten times the area of those 
of one cell. Therefore, by § 427, the internal resistance is 
only one tenth as much, i.e. 3 -^ 10, or 0.3 ohm. The equa- 
tion for the parallel arrang^ement becomes C = —--^ , 

^ ^ 0.2 + .3 

or 3 amperes, which is about six times as much as the 
series arrangement would produce. Hence for n similar 
cells joined in parallel. Ohm's law is 

E. M. F. of one cell E 

P resistance of one cell ' r _i_ I 
number of cells n 



EXERCISES 

The pupil should represent each of the following cases diagram- 
matically. 

1. In the discussion of the two methods of combining cells, for 
what kind of circuits is the series arrangement of cells shown to be 
suitable ? 

2. How much current will a dry cell of 2 ohms resistance and 
1.43 volts send through a wire of 25 ohms resistance? 

3. How much current would three cells similar to the one in 
Exer. 1 send through the same wire (1) when joined in series and 
(2) in parallel? Ans. (1) 0.13 ampere ; (2) 0.0557 ampere. 

4. What current would 8 Daniell cells of which the E. M. F. is 
1.08 volts and the resistance 3 ohms each send through an ammeter of 
0.4 ohm, when joined in series? What would be the current from 
one cell alone? 



ELECTRICAL MEASUREMENTS 413 

5. What would be the current produced from the same 8 cells 
connected in parallel and using the same ammeter ? 

6. Which is the better arrangement of 4 Daniell cells (E. M. F. 
= 1.1 volts and r = 2.5 ohms each) when the external resistance is 
6 ohms? 

7. Show that a small Daniell cell will give practically as much 
current as a large one through 1000 ohms of resistance, the resistance 
of the small one being 30 ohms while that of the large one is 4 ohms. 

8. If a galvanometer gives the same deflection when connected 
with a very small cell as it does when connected with one several 
times as large but having the same E. M. F., is the resistance of the 
instrument large or small ? 

9. A dry cell whose E. M. F. is 1.5 volts produces a current of 
0.2 ampere through an instrument whose resistance is 7 ohms. Find 
the resistance in the cell. 

10. Which will produce the greater effect in an external circuit of 
5 ohms, a dry cell whose resistance is 3 ohms or a copper-oxide cell 
(E. M. F. = 0.8 volt) having a resistance of 0.2 ohm? 

11. What current will be derived from a Daniell cell whose resist- 
ance is 3 ohms and a dry cell with a resistance of 2 ohms when they 
are joined in series and the external resistance is 2.74 ohms ? 

12. What would be the value of the current in the preceding 
exercise if the poles of the Daniell cell were set in opposition to those 
of the dry cell? Diagram the connections. 

13. A circuit contains 4 dry cells (E.M. F. of each = 1.5 volts) 
joined in series. Three of the cells are known to have a resistance 
of 0.5 ohm each, and the external resistance is 10 ohms. If the 
current is 0.15 ampere, what is the resistance offered by the fourth 
cell ? Would the current be increased or decreased by removing this 
cell from the circuit? 



2. ELECTRICAL ENERGY AND POWER 

435. Energy of an Electric Current. — We have learned 
under the study of the effects of #electricity (§§ 411, 413, 
414) that the energy of a current may be expended in 
three ways: (1) it may produce mechanical motion, as in 
the electric bell ; (2) it may produce chemical separation ; 
and (3) it may be converted into heat in a conductor. 



414 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. 351. — Electrical Energy is Trans- 
formed into Heat between A and B. 



Let US imagine an electric circuit as shown in Fig. 351. 
Between the points A and ^ is a resistance of 4 ohms. 
If the battery produces a current of 5 amperes, for ex- 
ample, the potential dif- 
ference between A and B 
will be, by Ohm's law 
(§ 431), 4x5, or 20 
volts. 

Now the work done, or 
energy expended, by the 
current between A and B depends on three factors, — 
(1) the potential difference, (2) the current strength, and 
(3) the time, — and it is measured by their product. 
Therefore we have the relation 

energy =^ potential difference x current strength x time. 

If the time is in seconds, the potential difference in 
volts, and the current strength in amperes, this product 
gives the energy in terms of a unit called the joule., 
which is equal to 10,000,000 ergs. Hence 

volts X amperes x seconds = joules. (6) 

In the electric circuit shown in Fig. 351 the amount of 
energy expended in 5 minutes (300 seconds) between the 
points A and B is, then, by equation (6), 20 x 5 x 300, or 
30,000 joules. 

436. Power of an Electric Current. — ^iwgq power refers 
to the rate at which work is done or energy expended 
(§ 57), it may be found by simply dividing the total 
energy expended by the time. In an electric circuit, 
therefore, the power is measured by the product of the 
potential difference and the current strength ; or, 

power = volts x amperes. (7) 

It is plain that power is the number of joules per second 
expended by the current. A power of one joule per 



ELECTRICAL MEASUREMENTS 415 

second is called a watt^ in honor of James Watt (1736- 
1819) of Scotland. One horse power is equal to 746 watts. 
In the example chosen in § 435, the power of the current 
in the circuit between A and ^ is 20 x 5, or 100 watts ; 
i.e. if the energy of the current which is expended be- 
tween the points A and B could be converted into mechani- 
cal energy, there would be developed ^||^ horse power. 

437. Quantity of Heat Developed by a Current. — When 
no mechanical or chemical work is done by a current of 
electricity, the energy is used simply in overcoming the 
resistance of the conductor and is converted into heat. 
Now one heat unit (a calorie) has been found by experi- 
ment to be equivalent to 4.2 joules; or, one joule equals 

— , or 0.24 calorie. 
4.2 

If we express the potential difference between two points 
of a circuit by E and the current strength by C, the num- 
ber of joules expended in t seconds is, by equation (6), 

XT 

EOt joules. But by Ohm's law (§ 431), (7= — ; whence 

E = OB. Now if we substitute this value of E, we obtain 
for the energy expended O^Bt^ which represents the amount 
of electrical energy that is converted into heat in a resist- 
ance of B ohms when the current is O amperes and the 
time t seconds. Reducing joules to calories gives 

Heat = 0.24 C^Rt calories. (8) 

This equation represents Joule's law, which may be 
stated as follows: 

The heat developed in a conductor hy an electric current is 
proportional to the square of the current^ to the resistance of 
the conductor^ and to the time the current is flowing. 

438. Loss of Energy in Transmitting Electricity. — The 
conversion of electrical energy into heat in a conductor 



416 A HIGH SCHOOL COURSE IN PHYSICS 

through which it flows has an important commercial bear- 
ing on the transmission of electric power over long lines. 
For example, if the resistance of the wires leading from a 
distant source of electrical power to a group of lamps is 
only 2 ohms, the waste of power in transmitting a current 
of 10 amperes is 10^ x 2, or 200 watts. Now 10 amperes 
at 110 volts would operate 20 lamps, each of which con- 
sumes 55 watts. The total power required would be, 
therefore, 20 x 55 -f 200, or 1300 watts. The loss in 
transmission is therefore y^- of the total power produced. 
Now if the number of lamps is increased to 100, the lamp 
consumption is 5500 watts, while the line loss (since the 
current is now 50 amperes) is 50^ x 2, or 5000 watts. 
Hence, of the 10,500 watts which must be produced at the 
power station, 5000 watts, or 47.6 per cent, are lost by 
being converted into heat in the line. 

The loss of energy in long lines of wire can be reduced 
by constructing the line of larger wire and thus decreasing 
the resistance factor, but in many cases this method is im- 
practicable from a commercial standpoint. However, by 
using modern devices involving principles that we shall 
study later, economical transmission of power over long 
distances is rendered possible. 

EXERCISES 

1. Compute the number of joules transmitted by a current of 10 
amperes maintained for 20 minutes at a potential difference of 110 
volts. 

2. Compute the heat loss per hour in an electric line of 3 ohms 
resistance when the current is 5 amperes. What is the result if the 
current is 10 amperes ? 

3. If the power required for an incandescent lamp is 60 watts, what 
is the consumption of 50 lamps measured in terms of the horse power? 

4. A piece of platinum wire is heated by a current of 2 amperes, 
and the potential difference between its ends is 8 volts. Compute the 
heat developed per minute. 



ELECTRICAL MEASUREMENTS 



417 



3. COMPUTATION AND MEASUREMENT OF RESISTANCES 

439. Resistances Computed. — It is customary to com- 
pute the resistance of a wire of given material, length, 
and size from the known resistance of a wii'e of that kind 
which is 1 foot in length and 0.001 inch (called 1 mil) 
in diameter. The following table gives this value in 
ohms for some of the common metals. 

Ohms of Resistance in Wires 1 Foot long and 0.001 Inch in 

Diameter 

Silver 9.5 Platinum 80 

Copper 10.2 German silver . . . . 180 

Iron 61.5 Mercury 570 

According to the laws of resistance stated in § 427, tJte 
resistance of a ivire can he calculated by simply multiplyiny 
the number given in the table by the length of the wire in feet 
and dividing by the square of the diameter, which must first 
be reduced to thousandths of an inch. For example, the 
resistance of a mile of iron wire 0.08 inch in diameter is 
61.5 ohms X 5280 -- 802, ^^ 59,7 ^^ms. 



EXERCISES 



1. What is the resistance of each of the wires given in the follow- 



ing table? 


















Kind of Wire . Number 


Diameter 


Lexgth 


Copper 8 


0.128 inch 


1 mile 


Copper . . . 












22 


0.025 inch 


500 feet 


Copper . . 












36 


0.005 inch 


40 feet 


Copper . . 












40 


0.003 inch 


25 feet 


Iron . . . 












9 


0.114 inch 


1 mile 


Iron . . . 












14 


0.064 inch 


25 feet 


Iron . . . 












. 10 


0.102 inch 


5000 feet 


German -silver 












. 20 


0.032 inch 


35 feet 


, German silver 








30 


0.010 inch 


10 meters 


Platinum 


» -> • 




24 


0.020 inch 


2 feet 


28 



















418 



A HIGH SCHOOL COURSE IN PHYSICS 




2. How long must a No. 36 copper wire be to offer a resistance of 
one ohm ? 

3. Find the diameter of a copper wire that has a resistance of 10 
ohms per thousand feet. 

4. A No. 22 wire 1000 ft. long offers a resistance of 285 ohms. 
Of what material mentioned in § 439 might the wire be made ? 

440. Resistance Boxes. — Coils of wire whose resistances 

have been carefully adjusted aud 
marked are incased in boxes and 
sold by electrical supply houses. 
Such a box is shown in Fig. 352. 
The ends of each coil are sol- 
dered to brass blocks A, B, C, 

Fig. 352. — A Resistance Box. etc., Fig. 353, between which 
brass plugs may be inserted. The blocks are mounted on 
an ebonite or hardwood plate which forms the cover of 
the box. When the plugs are all in place, 
no resistance is encountered by an electric 
current in passing from one block to the 
next ; but when a plug is withdrawn, the 
current must pass through the correspond- 
ing coil whose resistance is marked on 
the top of the box. Thus, by removing 
plugs, any resistance from the smallest up to the sum of 
all the resistances in the box can be obtained. 

441. Resistance Measured by Substitution. — Place a coil 

of wire whose resistance is to be determined in circuit with a Daniell 
cell and a galvanometer and read the deflection produced by the current. 
Remove the coil from the circuit and insert a resistance box. With- 
draw plugs from the box until the deflection of the galvanometer is 
the same as before. The sum of the resistances corresponding to the 
plugs that have been removed gives the resistance of the coil. Why? 

442. Resistances Measured by Voltmeter and Ammeter. — 

The coil of wire C, Fig. 347, whose resistance is to be found, is placed 
in a circuit with the cell B and an ammeter. A voltmeter of high 




Fig. 353. — Showing 
Arrangement of 
Coils. 



ELECTRICAL MEASUREMENTS 



419 




Fig. 354. — Diagram of Wheatstone's 
Bridge. 



resistance is joined to the terminals of the coil C to indicate the 

potential difference at those points. Since by Ohm's law (§ 431) 

the current through C equals E 

the potential difference in volts 

divided by the resistance in 

ohms (Eq. 2, § 431), the value 

of C can be found by dividing 

the reading of the voltmeter 

by that of the ammeter. It is 

essential that the resistance of 

the voltmeter be so large that 

practically none of the current 

can pass through it. 

443. The Wheatstone Bridge. — When the current from a cell 
B, Fig. 354, passes through the point A, it divides into two parts, the 
portion C flowing through the resistance R ohms, and the portion C 
through the resistance R' ohms in the straight uniform wire AD. 
Now a sensitive galvanometer G joining points E and F will be 
deflected unless E and F do not differ in potential. The point F which 
has the same potential as E is found by moving the contact along 
the wire AD until no deflection can be observed. 

Under these conditions the difference of potential between .1 and 
E must equal that between A and F. By Ohm's law (§ 431), 

fall of potential -^ resistance = current. 
Hence fall of potential — current x resistance^ 

whence CR = C'R'. (l) 

Again, the same current flows through X as through R, since the 
galvanometer is not deflected ; and the same current flows through 
R" as though R' for the same reason, Now, since the difference of 
potential between E and D is the same as that between F and D, we 

^^^« CX = C'R". (2) 

Dividing (2) by (1) we have 



X R" V j?R" 
— = — , or X = K — . 

R R' R' 

Now the resistances R' and R" are proportional to the lengths of 
the wire AF and FD (§ 427). Therefore, 



X=R 



FD 
FA* 



(3) 



420 A HIGH SCHOOL COURSE IN PHYSICS 

Four resistances combined in this manner constitute Wheatstone's 
Bridge. It is obvious that if the resistance R and the lengths of the 
wire AF and FD are known, the vaUie of the unknown resistance X 
can be calculated. The point F must be found experimentally as 
described above. 

444. Conductors in Series and Parallel. — Conductors 

are said to be joined in series when they are connected in 

^ A succession as shown in 

-^^^mmy----^mmi^^^^>mm^^ Fig. 355. in this con- 

FiG. 355. -Conductors Joined in Series, ^ition the entire current 
passes through each conductor. The combined resistance 
between A and B is the sum of the resistances of the sev- 
eral parts. 

The case is very different when the conductors are 
joined as shown in Fig. 356. Resistances combined in 
this manner are said to 
be connected in parallel. 
The most important com- 
bination of this kind is 
that of two wires thus con- 
nected. The current from 
the cell O will obviously 

divide into two parts at Fig. 356. -Conductors Joined 

point A which reunite at in Parallel. 

B. When the two resistances are equal, the two parts of 
the current are, of course, equal. But if the resistances 
are, for example, 3 and 7 ohms respectively, and the poten- 
tial difference between A and B common to both branches 
is 1 volt, the current through the upper branch is ^ am- 
pere and that through the lower one ^ ampere. Hence, 
while the resistances are as 3 is to 7, the currents are as 7 
is to 3. The currents from the cell, therefore, in a two- 
branched circuit are inversely proportional to the correspond- 
ing resistances. 




ELECTRICAL MEASUREMENTS 421 

Again, the total current flowing between A and B 
is ^ -h ^ ampere. If, now, we let M be the combined 

resistance between A and B, the total current is — ampere. 

Therefore i = ^ + i, (4) 

R 3 7 ^ ^ 

3x7 
whence B = , or 2.1 ohms. It is clear from the 

3 + 7 

example chosen, that the combined resistance offered hy the 
two branches of a divided electric circuit is the product of 
the two resistances divided by their su7n ; or, expressed as 

an equation, R = -^^ -• (5) 

^1 + 1*2 

445. Shunts. — A shunt is a conductor which is con- 
nected in an electric circuit parallel to another conductor. 
It may be likened to a side-track, or to a by-pass that is 
frequently used in the case of 
the conduction of water or gas 
through pipes. The term is 
applied to a resistance coil aS', 




Fig. 357, which is joined in a 
circuit parallel to a galvanom- 
eter G^ or some other instru- 

c\ 
ment. In this case it is used Fig. 357. — Illustiating the Use of 

to reduce the flow of electricity a Shunt, 

through the instrument; for, as shown in § 444, the cur- 
rent divides into parts that are inversely as the two resist- 
ances. Suppose, for example, the resistance of G- is 45 
ohms and that of aS' is 5 ohms. Then -^^ of the entire cur- 
rent passes through the galvanometer and ^^ of it through 
the shunt. Thus it is clear that when a shunt contains ^ as 
much resistance as a galvanometer, ^^ of the total current 
passes through the galvanometer, and -^-^ through the 



422 A HIGH SCHOOL COURSE IN PHYSICS 

shunt. In the same manner the division of a current in 
any given case may be computed. 

EXERCISES 

The pupil should diagram the conditions expressed in each of the 
following exercises. 

1. A current of 4 amperes divides and passes through two parallel 
coils of wire, one of 5 ohms and the other of 8 ohms. What is the 
current that goes through each branch ? 

Ans. 2.461 amperes and 1.539 amperes. 

2. Find the combined resistance offered by the two parallel con- 
ductors in Exer. 1. 

3. Two instruments of 3 and 4 ohms respectively are connected 
parallel between the poles of a dry cell (E.M. F. 1.5 volts) whose re- 
sistance is 1 ohm. Find (1) the external resistance of the circuit, 
(2) the current strength of the cell, and (3) the current flowing 
through each branch. 

Ans. (1) 1.714 ohms; (2) 0.553 ampere; (3) 0.316 ampere, 
0.237 ampere. 

4. Two electro-magnets of 4 and 12 ohms respectively are connected 
in parallel to each other and then placed in a circuit containing a 
coil of 3 ohms and a battery of 10 ohms. Find (1) the total resist- 
ance of the circuit, and (2) the strength of the current in each part 
of the circuit, the E. M. F. being 4.8 volts. 

Aiis. (1) 16 ohms; (2) 0.225 ampere and 0.075 ampere. 

5. A galvanometer of 30 ohms has a shunt of 30 ohms. When 
connected in an electric circuit, what part of the whole current will 
pass through the instrument? By what number must the cur- 
rent measured by the galvanometer be multiplied to give the entire 
current ? 

6. If the resistance of the shunt in Exer. 5 is reduced to 20 ohms, 
what part of the total current will the galvanometer carry, and what 
will be the multiplier? Ans. The multiplier will be 2.5. 

7. If a galvanometer has 300 ohms of resistance, what must be the 
resistance of a shunt so that only one tenth of the entire current will 
pass through the wire of the instrument? Ans. 33^ ohms. 

SUMMARY 

1. The unit of current strength is the ampere. It is 
the current that will deposit 0.001118 g. of silver per 



ELECTRICAL MEASUREMENTS 423 

second. Current strength is measured by an ammeter 
(§ 422). 

2. The electrical resistance of cylindrical wires of 
uniform size and composition is directly proportional to 
the length and inversely proportional to the square of tlie 
diameter. It also varies with the nature of the substance 
of which it is composed. The unit of resistance is the 
ohm (§§ 427 and 428). 

3. The unit of potential difference is the volt. It is 
the potential difference required to force a current of an 
ampere through a resistance of one ohm. Potential dif- 
ferences are measured by the voltmeter (§§ 429 and 430). 

4. Current, potential difference, and resistance are re- 

rr 

lated mathematically as shown by the equation (7=—, or 

amperes — volts -i- ohms. This relation is known as Ohm's 
Law (§ 431). 

5. The resistance of circuits includes interjial and 
external resistances ; the former is the resistance offered 
by the cells, the latter, that of the remaining portion of 
the circuit (§ 432). 

6. For n similar cells joined in series the current is 

K + nr 

7. For n similar cells joined in parallel the current is 

C= -^— (§ 434). 

n 

8. The energy expended in any part of a circuit is the 
product of the potential difference, current strength, and 
time, or 

energy — ECt joules (§ 435). 



424 A HIGH SCHOOL COURSE IN PHYSICS 

9. The power, or rate at which the current is working, 
is power = EO watts (§ 436). 

10. The quantity of heat developed by a current of 
Q amperes in a resistance R ohms in t seconds is 

heat = 0.24 C^Rt calories (§ 437). 

11. The resistance of a cylindrical wire may be com- 
puted by multiplying the resistance of one foot of it one 
mil in diameter by the whole length and dividing by the 
square of the diameter in mils (§ 439). 

12. The combined resistance of conductors joined in 
series is their sum. The combined resistance of two con- 
ductors joined in parallel is 

R=Jhr!i^ (§ 444). 
^1 + r^ 



CHAPTER XX 



ELECTRO-MAGNETIC INDUCTION 



1. INDUCED CURRENTS OF ELECTRICITY 

446. Currents Induced by Magnetism. — Oersted's dis- 
covery of the effect of a current of electricity upon a mag- 
netic needle (§ 400) in 1819 led to the invention of the 
electro-magnet by Sturgeon in 1825, and induced many 
experimenters to seek for a method of producing an elec- 
tric current by means of a magnet. Two physicists, 
Joseph Henry 1 in America and Michael Faraday ^ m Eng- 
land, independently discovered the process for doing this 
about 1831. 

Connect the ends of a coil of insulated wire C, Fig. 358, consisting 
of a large number of turns, directly to the termi- 
nals of a sensitive d'Arsonval galvanometer. Ar- 
range a large horseshoe magnet with its poles 
upward as shown. Now move the coil down quickly 
into the magnetic field. The galvanometer will re- 
veal the presence of a current of electricity, hut the 
index will go hack to zero as soon as the coil stops mov- 
ing. If the coil be now removed from the magnetic 
field, the galvanometer will show that a current is 
produced in the opposite direction. Repeat the ex- 
periment, but move the coil more slowly. Turn the 
coil over and repeat the experiment. Each deflec- 
tion will be in a direction opposite to the corres- 
ponding one produced at first. Fig. 358. — Induc- 
ing an Electric 

The experiment shows clearly the produc- Current. 
tion of an electric current without the aid of a voltaic cell. 

1 See portraits facing page 426. See also Maxwell, facing page 432. 

425 




426 A HIGH SCHOOL COURSE IN PHYSICS 

Such a current iS called an induced current. The experi- 
ment shows also that the induced current flows only while 
the coil is moving in the magnetic field, i.e. only while the 
number of lines of force through the coil is changing. 
Since a current is always due to an E. M. F., it is obvious 
that the movement of a coil of wire in the magnetic field 
develops such an electromotive force in the wire. This 
is called an induced electromotive force. Furthermore, the 
slower the movements, tlie less rapid the change, and the 
smaller the induced E. M. F. The following general laws 
may be stated : 

1. A change in the number of magnetic lines of force 
threading through a coil (or a single loop) of wire induces 
an electromotive force in that coil. 

2. The induced electromotive force is proportional to the 
rate at which the number of lines of force is changed. 

447. Special Cases of Current Induction. — According to 
the laws of induction given in the preceding section, an 
induced electromotive force may be brought about either 
by an increase or by a decrease in the number of magnetic 
lines through a coil of wire. This effect may be produced 
in several different ways, as the following experiments 
will show. 

1. Connect the ends of a coil of wire with a sensitive galvanometer 
and thrust the N-pole of a magnet into the coil. A deflection will be 
produced. On pulling the same pole out of the coil, a deflection in 
the opposite direction results. Turn the coil over and repeat the 
operations. Every effect is just the reverse of the corresponding one 
produced at first. The experiment should be repeated, using the S- 
pole in the same manner. 

2. Connect a coil of wire. Fig. 359, to the poles of a voltaic cell 
and thrust this coil into the coil used in Experiment 1. In every 
operation similar to those performed above, the effect is the same as 
that produced by using a magnet. Repeat with a soft iron core within 
the smaller coil. 



r" 



MICHAEL FARADAY (1791-1867) 

Faraday was one of the most distin- 
guished chemists and physicists of the 
nineteenth century. He was the son of 
a blacksmith at Newington, near London, 
England, and became a bookbinder in 1804. 
Hearing by chance some lectures on chem- 
istry by Sir Humphry Davy of the Royal 
Institution, he became interested in the 
subject, and in 1813 was appointed assist- 
ant in Davy's laboratory. In 1825 he be- 
came director of this laboratory, and in 
1833 was made professor of chemistry in 
the Royal Institution for life. He died at 
Hampton Court in 1867. 

Faraday's achievements in the domain 
of chemistry are largely overshadowed by his numerous and brilliant 
discoveries in electricity, of which the most far-reaching was that 
of the induction of electric currents by magnets, made in 1831. This 
subject has led to results of tremendous value in the commercial 
applications of electricity. 

His further discoveries deal with the capacity of condensers, the 
laws of electrolysis, the rotation of the plane of polarized light by 
a magnetic field, and diamagnetism. 

Faraday was a prolific writer and a popular lecturer on scien- 
tific subjects. He is best known by his Experimental Researches 
in Chemistry and Physics. 




JOSEPH HENRY (1797-1878) 



After the time of Franklin, Henry was 
the first in the United States to make orig- 
inal researches in the subject of electricity. 
After the invention of a practical electro- 
magnet by Sturgeon in 18:35, Henry devel- 
oped a magnet, in the construction of which 
he employed many turns of copper wire in- 
sulated with silk. The magnet was capable 
of supporting over fifty times its own weight. 
It is believed on good evidence by many 
physicists that the discovery of electro- 
magnetic induction was made by Henry at 
Albany, New York, in 1830, although he did 
not publish his results until 1832, a year later 
than the publication of Faraday's results. 




ELECTRO-MAGNETIC INDUCTION 



427 



3. Open tlie battery circuit used in Experiment 2, insert a key, and 
place one coil within the other. Complete the battery circuit by press- 
ing the key. A deflection 
of the galvanometer will 
indicate the presence of an 
induced current which at 
once dies away. Open the 
key, and an induced cur- 
rent in the opposite direc- 
tion wdll be obtained. 

4. If the galvanometer 
is sufficiently sensitive, sim- 
ply turning the coil with which it is connected in the earth's magnetic 
field will suffice to produce an appreciable current. 




Fig. 359. — Inducing a Current by a Current. 



It is to be observed that in each of the cases chosen in 
the experiments, a change in the number of magnetic lines 
through the coil is brought about in the process. In Ex- 
periment 1, the magnet carries its lines of force with it 
when it is thrust into the coil ; in Experiment 2, the mag- 
netic lines of force set up by the battery current in one 
coil are carried through the second coil when the former 
is thrust into the latter ; in Experiment 3, the change in 
the number of lines is produced by alternately making 
and breaking the circuit which contains the battery. In 

every case the induced current 
Induced — jy set up by magnetic action, i.e. 
Cinien_ qy\\\qv by an increase or a decrease 

in the number of magnetic lines 

through the coil connected with 

the galvanometer. 




448. Direction of the Induced 
Current. — Lanzas Law. — Connect 

a small piece of zinc with one terminal 
of a sensitive galvanometer and a piece 

Fig. 3(30. ~ Showing the Direc- ^^ ^"^PP^^' ^^i*^ the other. Hold the two 
tion of the Induced Current, metals in the fingers aijd observe the 



428 A HIGH SCHOOL COURSE IN PHYSICS 

deflection of the galvanometer. The moisture of the hand is sufficient 
to establish a current from the copper to the zinc through the instru- 
ment. Now repeat the experiment of § 446, and from the direction 
in which the galvanometer is deflected, determine the direction of the 
induced current in the wire. 

While the coil is moving into the magnetic field placed 
as shown in Fig. 360, the current in the wire will be 
found to have the direction shown by the arrows. Thus 
the induced current tends to set up a magnetic field of its 
own whose lines of force are opposite in direction to those 
of the magnet. This fact is easily verified by applying the 

rule given in § 408. See Fig. 
361. Hence, increasing the mag- 
netie lines through the coil produces a 
current which tends to oppose that in- 
crease. On the other hand, moving 
the coil away from the magnet in- 
duces in the coil a current in the 
direction opposite to the former cur- 
FiG. 361. — Applying rent. Heucc, a decrease in the num- 
ber of lines through the coil sets up a 
current that tends to prevent that decrease in number. In 
fact, every case of current induction may be shown to obey 
the following law: 

The direction of an induced current is such as to produce 
a magnetic field that will tend to prevent a change in the 
number of magnetic lines of force through the coil. 

This is known as Lenz*s Law and may be viewed as an 
application of the more general law of the Conservation of 
Energy (§ 64). From this law we know that neither elec- 
trical nor any other form of energy can be derived unless 
an equivalent amount of work be performed. In this 
case the work is done when the coil is forced into the 




ELECTRO-MAGNETIC INDUCTION 



429 



-• » 




Fig. 362. 



Diagram Showing the !Parts of 
an Induction Coil. 



magnetic field in opposition to the repulsion between the 
field of the magnet and that of the induced current. 

449. The Induction Coil. — One of the most important 
applications of the principles of electro-magnetic induction 
is found in the induction 
coil. The instrument 
consists of a so-called 
primary coil of coarse 
wire A^ Fig. 362, wound 
on a core of soft iron, a 
secondary coil of several 
thousand turns of very 
fine insulated wire S^ 
and a current interrupter 
I, The terminals of the 
secondary coil are at p 
and q. 

When the current from a battery B flows through the 
primary coil, it magnetizes the iron core. The iron ham- 
mer a is then drawn toward the core and away from the 
screw h. This operation breaks the primary circuit at the 
screw point d. The interruption of the current at d causes 
the core to lose its magnetism and release the hammer «, 
which is restored to its original position by the spring to 
which it is attached. The contact at d is thus made again, 
and all the operations are repeated. It is clear that most 
of the lines of force set up by the battery current in the 
primary coil pass through each turn of the secondary. 
Hence, when the current is interrupted and the lines of 
force decrease, an E. M.F. is induced in the secondary coil. 
Even with small coils, the potential difference between p 
and q is often sufficient to cause a spark to pass across a 
short gap from one to the other. Again, when the primary 
circuit is made at c?, the resulting increase in the number 



430 A HIGH SCHOOL COURSE IN PHYSICS 

of lines of force induces a contrary E. M. F. in the second- 
ary, but this is of a much smaller intensity than the former 
induced E. M. F. Lenz's law applied to the induction 
coil shows the following statements to hold true : 

1. When a current is started in the primary coil^ a momen- 
tary current is induced in the secondary in the opposite 
direction. 

2. Whe7i the current in the primary coil is interrupted^ a 
momentary current is induced in the secondary m the same 
direction as that in the primary. 

The condenser is introduced as an accessory part of the 
induction coil in order to increase the rate of demagnetiza- 
tion of the iron core. By its aid a very quick interruption 

of the current in the 
primary coil is ac- 
complished, and 
consequently the in- 
fe: duced E. M. F. is 
proportionately in- 
creased. The use 
of an extremely large number of turns of wire in the 
secondary coil makes it possible to secure a sufficiently 
large potential difference between p and q to produce 
sparks many inches in length. The actual form of the in- 
strument is shown in Fig. 363. 

If metallic handles that can be grasped with the hands are joined 
to the terminals of the secondary coil, it will be found that the 
shock experienced when the primary circuit is made is much lighter 
than at the break. When the primary circuit is made, the current 
requires a large fraction of a second to attain its maximum value ; 
consequently the rate of change of the lines of force is comparatively 
small. At the break of the primary, however, the current falls to zero 
very suddenly, thus producing a high rate of decrease in the lines of 
force. I'or this reason (§ 446) the E,M. F. is correspondingly greater 



Fig. 363. — An Induction Coil. 



ELECTRO-MAGNETIC INDUCTION 



431 



at the interruption of the primary than at the making of the 
circuit. 

The induction coil is widely used in many operations. It is used 
in gasoline engines to ignite the mixture of air and vapor (§271), 
for igniting explosives in blasting, and in chemical laboratories for 
many experimental purposes. Small induction coils are used in medi- 
cine for the remedial effect of the electric current and in the speaking 
part (transmitter circuit) of the modern telephone. Coils of large 
dimensions are employed in sending messages by wireless telegraphy 
and in the production of X-rays. 

450. An Induced E.M.F. in a Moving Conductor. — By 

referring to Fig. 360 it becomes clear that in order to 
change the number of lines of force threading through the 
coil of wire, the conductor has to cut through, or across, 
some of these lines. The following experiment will show 
the effect of this operation in another way. 

Hang a loose copper wire across the classroom and connect its ends 
with a sensitive galvanometer. If the wire be set swinging, the galva- 
nometer will be deflected first to the right and then to the left as the 
conductor cuts through the earth's lines of force. Connect one of the 
galvanometer wires to the center of the swinging conductor and repeat 
the experiment. The induced E. M. F. will be about one half as great 
as before, as shown by the reduced deflection of the galvanometer. 

The experiment shows th;it when a conductor cuts across 
magnetic lines of force., an E.M.F. is induced within it. 
If, for example, the con- 
ductor ah, Fig. 364, 
move from right to left, 
and the galvanometer be 
used to determine the 
direction of the E.M. F., 
it will be found that the 
E. M. F. acts through 
the conductor from a 

towards b ; i.e. b will 

Fig. 3(54. — A Conductor ah Cutting Mag- 
act temporarily like the netic Lines of Force. 




432 A HIGH SCHOOL COURSE IN PHYSICS 

positive pole of a cell, and a like the negative pole. While 
the wire is swinging in the opposite direction, the lines 
of force are cut from the opposite side, and the induced 
E. M. F. is reversed. In the second part of the experi- 
ment the lines of force are cut only one half as fast, and 
the reduced deflection shows that the value of the E. M. F. 
varies with the rate at which the lines of force are cut by 
the moving conductor. 

It is clear that the direction of the induced E. M.F. (or 
current) depends (1) on the direction of the lines of force 
and (2) on the direction of the motion of the conductor. 
The three quantities, viz. Motion^ Force^ and Current^ have 
a definite relation which has given rise to the following 
rule: 

Let the thumb and the first ttvo fingers of the right hand he 

bent at right angles to each other, as 

in Fig. 365. If now, the thumb 

point in the direction of the Motion 

of the wire, and the First finger 

point in the direction of the lines of 

force in the Field, then the Center 

fi7iger indicates the direction of the 
Fig. 365— Finding the Di- r„^,.g«t 
rection of an Induced Cur- 

^^" ■ The key to the rule is found in 

tlie corresponding initial letters in the words First and 
Field, Center and Current. 

2. DYNAMO-ELECTRIC MACHINERY 

451. Principle of the Dynamo. — The laws of induced 
currents find their most important application in the 
dynamo, a machine which facilitates the conversion of 
mechanical energy into electrical. The familiar examples 
of electric street lighting and the operation of city and 





JAMES CLERK-MAXWELL (1831-1879) 



Maxwell ranks as one of the greatest of mathematical physicists 
on account of the important practical results to which his theories 
have led. He was born in Edinburgh, Scotland, where he received 
his early education. In 1856 he became professor of physics in 
Marischal College at Aberdeen, in 1860 professor of physics and 
astronomy in King's College of London, and in 1871 first pro- 
fessor of physics in Cambridge University. 

Maxwell built upon the experimental discoveries of Faraday, so 
arranging and relating them as to make them yield to mathematical 
treatment. He advocated the view that electric and magnetic forces 
result from certain changes in the distribution of energy in the 
ether. He showed that electro-magnetic action must travel through 
space in the form of transverse waves and with the velocity of light. 
Later on this theory was corroborated by the illustrations of Hertz, 
who was first to produce such waves and show that they could be 
reflected, refracted, and polarized like light. By these two physicists, 
the one attacking problems from the mathematical standpoint, the 
other building his experiments upon the theoretical results obtained, 
the intimate relation between light and electricity has been amply 
confirmed. 

Among other subjects investigated by Maxwell was the kinetic 
theory of gases. The results were published with others in a 
memorial edition of two volumes by the Cambridge Press. 

Maxwell's works also include his Theory of Heat (1871), EUc- 
tricity and Magnetism (1873), and a clear and concise treatise of 
dynamics entitled Matter and Motion (1876). 



ELECTRO-MAGNETIC INDUCTION 



433 



Wires to Galvanometer 




Fig. 3(>(). — Illustrating the 
Dynamo Principle. 



interurban railroads have been brought about by the devel- 
opment of the dynamo. 

Make a rectangular coil of 200 or 300 turns of very small copper 
wire of such dimensions as to rotate between the poles of a horse- 
shoe magnet. Connect the ends with a 
d'Arsonval galvanometer and place the coil 
in the magnetic field as shown in Fig. 366. 
Starting with the plane of the coil at right 
angles to the lines of force, i.e. vertical, ro- 
tate it through 90.° A deflection will show 
the presence of an induced current. Con- 
tinue the rotation through the next 90°. A 
deflection in the same direction will be ob- 
served. If, now, the rotation be continued, 
a deflection in the opposite direction will 
be produced until the coil has returned to 
its initial position. 

When the coil is revolved from 
the position shown in Fig. 367, the two portions a and b 
begin to cut the lines of force of the magnet and thus de- 
crease the number of lines passing through 
the coil. A current is ther^jfore set up 
by the induced E, M. F., which, accord- 
ing to the rule given in the preceding 
section, is upward in a and downward in 

h. P^or the first 180° of rotation, the por- 

<X^^^ tion a of the coil continues to cut the 

Fig. 3G7. — Loops of fines of force in the same direction ; and 

Wire P6rD6]iclici.i~ 

lar to Lines of the same is true of portion b; hence the 
^°^^®- induced current thus far is continuous 

and in one direction. When, however, the coil begins to 
revolve through the second 180°, each part of it begins 
to cut magnetic lines in the opposite direction; hence the 
induced current will accordingly flow through the wire in 
the opposite direction. It is now obvious that a continuous 
rotation of the coil develops a current whose direction reverses 
29 




434 



A HIGH SCHOOL COURSE IN PHYSICS 




Fig. ;^08. — Diagram of a Simple 
Alternating-current Dynamo. 



every time the plane of the coil is perpendicular to the lines 
of force. Such a current is called an alternating current. 
452. The Alternating-current Dynamo. — Alternating- 
current dynamos, or alternators., are based upon the prin- 
X ciples of induction as shown by 

the experiment in the preceding 
section. In its simplest form, 
the machine consists of a coil of 
several turns of wire arranged 
to rotate between the poles of a 
strong electro-magnet, as shown 
in Fig. 368. The wire is wound 
on a core of soft iron C and 
mounted on a shaft A. The re- 
volving coil is called the armature, and the electro-magnet 
whose poles are N and S is called the field magnet. 
In order to provide for the flow of the induced current to 
and from the rotating coil, the ends of the wire of the ar- 
mature are connected wdth the two insulated metal collect- 
ing rings a and a' . Stationary '' brushes," ^, 5, rest against 
these rings as shown and conduct the induced current to 
the external circuit X. The efficiency of the machine is 
greatly increased by the presence of the iron core (7, since 
it multiplies and concentrates the number of magnetic 
lines of force between the poles N and S. 

When the armature is rotated, an induced alternating 
current is set up in the coil, and through the rings and 
brushes is transmitted to the external circuit. The E. 
M. F. will be determined by the number of lines of force 
cut per second. This will of course be greatest when the 
coils are moving at right angles to the magnetic lines., i.e. 
when the plane of the coils is horizontal. The E. M. F. 
is zero at the instant of reversal, for at that time all por- 
tions of the coil are moving parallel to the lines of force. 



ELECTRO-MAGNETIC INDUCTION 



435 




Fig. 369. — Diagram of an Alternating 
Current. 



Hence, the E. M. F. rises from zero to a maximum value 
and then decreases to zero again, at which time it reverses. 
Such a current is shown 
diagrammatically by the 
curve in Fig. 369. 

453. Multipolar Ma- 
chines. — Alternators are 
of great commercial value in generating electricity for both 
lighting and power. They have largely replaced the so- 
called direct-current dy- 
namos which have had a 
world-wide use. But for 
practical purposes it is 
desirable that the rever- 
sals of an alternating 
current attain, or even 
exceed, 120 per second. 
This, of course, cannot be 
accomplished with the 

two-pole machine described in the preceding section. In 
the multipolar machine there are several poles arranged 
around the armature as shown in Fig. 370. In this ma- 
chine there are as many coils of wire in the armature as 
there are poles in the field magnets. 

As the moving coils cut through the lines of force, an 
induced E. M. F. is set up in each. The several coils are 
so connected that the E. M. F. at the brushes is the sum 
of that produced in the separate coils. The field magnets 
are excited by means of a continuous direct current from 
a small dynamo (§ 454). Figure 371 shows a type of 
multipolar alternator which is in common use. 

454. The Direct-current Dynamo. — The alternating-cur- 
rent dynamo described in § 452 may be employed to pro- 
duce a unidirectional current by the introduction of a 




Fig. 370. 



Diagram of a Multipolar 
Machine. 



436 



A HIGH SCHOOL COURSE IN PHYSICS 



so-called commutator. The commutator in this case con- 
sists of a metal ring divided into two semicircular parts 




Fig. 371. — A Multipolar Alternator. 

a and a\ Fig. 372, called segments. These are insulated 
from each other and mounted on the shaft which car- 
ries the armature. Each of the two ends of the arma- 
ture coil connects with a seg- 
ment of the commutator as 
shown in the figure. The 
brushes h and 5' which conduct 
the induced current to and from 
the external circuit X are set on 
opposite sides of the commuta- 
FiG. 372. — Diagram of a Direct- tor. Their position is very im- 
current Dynamo. portant. They must be SO placed 

that each brush changes its point of contact from one segment 
to the other at the instant the current reverses in the armature 
coil. Thus 5, for example, will continually be in contact 




ELECTRO-MAGNETIC INDUCTION 



437 




Fig. 373. — (1) Diagram of the Armature 
Current. (2) Diagram of the Current 
Taken from the Brushes. 



with a positively charged segment, and b' with a negatively 
charged one. Hence, in this case, the current will always 
flow into the external 
circuit through h and 
return to the armature 
through b'. Such a 
current is pulsating in 
nature, since the 
E. M. F. falls to zero 
twice during each revo- 
lution (§ 452). A comparison of the alternating current 
in the armature with the pulsating current taken off at the 
brushes is made in Fig. 373. 

455. Dynamos for Steady Currents. — The pulsating cur- 
rents produced by dynamos of one coil (Fig. 373) are un- 
satisfactory for most purposes. The difficulty may be 
overcome and a continuous current developed by the use 
of several coils distributed uniformly over the armature. 

The first armature 
wound in this manner 
was the so-called 
Gramme ring, invented 
by a Frenchman in 1870. 
The core of the arma- 
ture is an iron ring so 
mounted on an axle as to 
turn between the poles, 
iVand S, Fig. 374, of a strong electro-magnet. The ring 
is wound with several coils of copper wire placed at equal 
distances. These are represented in the figure by the sin- 
gle turns numbered from 1 to 12. At each junction of 
two adjacent coils a connection is made with a segment of 
the commutator (7, which consists of as many insulated 
bars as there are coils in the armature. The lines of force 



From the External 




Circuit 

Fig 374. — Diagram of the Gramme 
Ring Armature. 



438 



A HIGH SCHOOL COURSE IN PHYSICS 



follow through the iron of the ring from iVto S, as shown 
by the dotted lines ; hence the outer portions of each coil 
are the only ones that cut the lines when the armature re- 
volves. 

if, now, the armature is revolved in the direction shown 
by the arrow at the top, the conductors from 1 to 5 cut 
through the lines in a downward direction ; hence the 
E.M.F. throughout these coils is in the direction shown 
by the arrowheads on the wire (§ 450). At the same 
time the conductors from 7 to 11 are cutting the lines in 
an upward direction, which develops an E.M.F. within 
them as shown by the affixed arrows. An inspection of 
the figure will show that upon both the right and the left 
side of the armature the tendency is to raise the lowest 
segment of the commutator to a high potential and to 
reduce the topmost one to a low potential. Therefore, 
by placing the brushes b and ¥ in contact with these 
points, a direct current is led through the external cir- 
cuit in the direction 
shown in the figure. The 
E. M. F. produced by such 
an armature is practically 
constant when the rate of 
rotation is uniform. 

456. The Drum Arma- 
ture. — The modern direct- 
current dynamo is pro- 
vided with an armature 
ture of the ''drum" type. 
This consists of a cylindrical core, or drum, of iron 
upon which are wound numerous coils of wire equally 
spaced. The construction is made clear by a study of 
Fig. 375. If the windings are traced, the coils will be 
found to be joined in series, and at four points connec- 




FiG. 375. 



Diagram of the Drum Arma- 
ture. 



ELECTRO-MAGNETIC INDUCTION 



439 



tions are made with the segments of the commutator C 
If the armature is now rotated between the poles of a 
magnet, the E. M. F. will at no time be zero at the 
brushes ; for at every instant some of the conductors are 
cutting magnetic lines. By setting the brushes at the 
proper points, a direct and fairly steady current will be 
transmitted to the external circuit. 

457. Field Magnets. — The magnetic poles between 
which the armature of a dynamo rotates receive their ex- 
citation from coils of wire carrying an electric current. 
In direct-current machines this current is produced by 
the dynamo itself. The initial current developed in the 
armature depends upon the residual magnetism (§ 379) 
of the pole pieces, which serves to induce a small current. 
This in turn increases the magnetism until the poles 
finally reach their full strength. 

In the series-wound dynamo, (1), Fig. 376, the entire 
current is led from the brushes through the few turns of 



Main Circuit 



Main Circuit 




(I) (2) (3) 

Fig. 376. — (i) Series- wound Dynamo. (2) Shunt-wound Dynamo. 
(5) Compound-wound Dynamo. 



thick wire of the field magnets, which are joined in series 
with the external circuit as shown. 

In the shunt-wound dynamo, (2), Fig. 376, only a portion 
of the entire current is led through the many turns of rather 



440 



A HIGH SCHOOL COURSE IN PHYSICS 



fine wire in the coils of the field magnet, while the main 
portion is conducted through the external circuit. The 
field magnet coils thus form a shunt (§ 445) to the ex- 
ternal circuit. 

In the compound-wound dynamo, (3), Fig. 376, the field 
magnets are wound with two coils, one being joined in 
series with the external circuit, and the other connected 
as a shunt between the brushes. A compound-wound 
machine adjusts itself to variations in the resistance of 
the external circuit in such a way as to maintain a con- 
stant difference of potential between the brushes. 

458. The Acyclic, or Unipolar, Generator. — It is of in- 
terest to note tliat the alternations of the current induced 
in the armature of a dynamo can be prevented by employ- 
ing a mechanism of the proper form. Figure 377 shows a 
sketch of a so-called acyclic^ or unipolar^ generator, which 




Fig. 377. — Diagram of an Acyclic Generator. 

consists of two collecting rings R and R' joined together 
by conducting bars i, ^, 3, etc. These bars are attached 
at their centers to a revolving shaft 2>, but are insulated 
from it. Magnets are placed with their poles as shown 
at iVand S. When the armature is revolved in the direc- 



ELECTRO-MAGNETIC INDUCTION 



441 



tion shown by the arrows MM'^ conductor 1 cuts through 
the field of the magnets, and an E. M. F. is induced in it 
(§ 450) in the direction shown by the arrow at C. Hence 
a current may be taken from the brush at B\ which 
rests continually against the collecting ring, through the 
external circuit X back to the brush B. Since each con- 
ductor cuts the magnetic lines of force in the same direc- 
tion, B' is always the positive brush and B the negative, 
and hence a direct current flows through both the inter- 
nal and external parts of the circuit. 

In the commercial form of this type. Fig. 378, the mag- 
netic field extends completely around the armature and is 




Fig. 378. — An Acyclic Generator. 

excited by means of a current in the field coils FF which 
lie in concentric circles around the cylindrical steel core 
A A. The armature conductors i, ^, 3^ etc., are attached 
to the periphery of a steel cylinder from which they are 
insulated. 

Unipolar generators are designed to run at a high speed, 
and a desired E. M. F. is obtained by arranging several 



442 A HIGH SCHOOL COURSE IN PHYSICS 

independent sets of conductors and their corresponding 
collecting rings in the armature of the machine and join- 
ing their brushes in series. In this manner the E. M. F.s 
of the separate sets are added together. The greatest 
advantages of this form of generator over that of the 
commutator type are the elimination of commutator diffi- 
culties and lower cost of construction. 

EXERCISES 

1. In order to show that a current of electricity is produced when 
a magnet is thrust into a coil of wire, why is it usually necessary to 
employ a coil of many turns? 

2. Account for the enormous difference of potential induced be- 
tween the terminals of the secondary of an induction coil ? 

3. Connect one terminal of a spark coil (induction coil) to the 
outer coating of a Ley den jar and bring the other close to the knob. 
Put the coil in operation and show that the jar becomes charged. 
Explain how the charge can "jump" into the jar, but cannot escape. 

4. How many revolutions per minute would have to be made by 
a two-pole alternator to produce 120 alternations of the current per 
second ? 

5. An alternator has 16 poles. How many alternations per second 
will be produced when the speed is 375 revolutions per minute? 

6. What will be the effect upon the E. M. F. of a shunt-wound 
dynamo of introducing resistance into the field magnet circuit? 

Suggestion. — Consider the effect of the added resistance on the 
current in the field coils and also on the magnetic field. 

7. Does this suggest a way in which the E. M. F. of a shunt-wound 
machine can be regulated ? 

3. TRANSFORMATION OF POWER AND ITS APPLICATIONS 

459. The Electric Motor. — Electrical energy is trans- 
formed into mechanical by means of electric motors. The 
direct-current motor does not differ greatly in construction 
from the dynamo. The principle underlying its action is 
illustrated by the following experiment. 



ELECTRO-MAGNETIC INDUCTION 



443 



Suspend a wire on the apparatus described in § 412 so that its 
lower end just dips into mercury. Hold a horseshoe magnet as 
shown in Fig. 379 and send a 
current from a new dry cell 
through the wire. The wire 
will be found to move at right 
angles to the lines of force, as 
shown by the arrow. If the cur- 
rent is nowreversed, the wire will 
move in the opposite direction. 







Movement of a Conductor in 
a Magnetic Field. 



The experiment shows 
clearly that a conductor in 
which a current is flotving 
tefids to move in a direction 
at right angles to the lines 
of force of a magnetic field fig. 379 
in which it is placed. The 
motion may be determmed by applying the rule of § 450, 
but by using the left hand instead of the right. 

Let this principle be applied to Fig. 374. If a current 
from some source be sent through the armature from h' to 5, 
the current through the coils will take the direction indi- 
cated by the arrows, dividing as it leaves h' . According to 
the principle shown in the experiment, all the conductors 
from i to 5 will be urged in an upward direction, and those 
from 7 to 11^ downward. Since the current employed by 
the motor flows through the field magnet coils, a powerful 
magnetic field is produced in which the armature will ro- 
tate with sufficient power to turn the wheels of factories 
and propel electric cars, launches, automobiles, etc. 

460. The Electric Railway Car. — A familiar applica- 
tion of the electric motor for the generation of mechanical 
power is found in the electric railway, Fig. 380. A 
current from the generator at the power house is trans- 
mitted through the trolley wire, or in some cases through 



444 



A HIGH SCHOOL COURSE IN PHYSICS 



a third rail, and by means of a metallic arm A to the 
motors placed under the floor of the car at M. The axles 
of the car are so geared to those of the motors that the 
car is propelled hy the rotation of the motor armatures. 

I 




Omnnnnn 



3; 



Track 




M 
Fig. 380. — Diagram of an Electric Railway System. 

From the motors the current returns to the power house 
through the track, the rails being carefully " bonded " 
together with copper conductors. In circuit with the 
motors is placed the controller (?, operated by the motor- 
man. By means of a series of resistance coils connected 
with the controller, the current can be increased or di- 
minished and the speed of the car thus regulated. An- 
other device enables the motorman to reverse the motors. 
A third accessory serves in applying the air brakes 
(§ 159) to check the speed after the current has been 
completely interrupted at the controller. 

461. The Alternating-current Transformer. — Alternat- 
ing currents owe their extensive application to the fact 
that the potential difference between two points can be 
easily reduced from a dangerous one of many thousand 
volts to one of a safe value for dwelling-house use and 
for many other purposes. This is accomplished by means 
of a transformer, which is simply a modified induction 
coil. 

The principle involved in the transformer is easily 
understood from a study of Fig. 381. An iron core M 
is wound with two independent coils of wire P and S. 



ELECTRO-MAGNETIC INDUCTION 



445 




Fig. 381. — Diagram of a Transformer. 



Let an alternating current be sent through the primary 
coil P, and let the secondary be connected with a group 
of lamps L. The cur- 
rent in P magnetizes 
the iron core in one 
direction, then demag- 
netizes and remagnet- 
izes it again in the 

opposite direction, while the magnetic lines follow the iron 
core through the secondary coil S. As a result of these 
magnetic changes in the core an alternating current is in- 
duced in S and flows through the lamps. 

If there are more turns of wire on the secondary coil 
S than on the primary coil P, the potential difference at 

the terminals of S will be 
greater than at P, because 
of the larger number of 
loops of wire in which the 
magnetic changes occur. 
In this case the transformer 
is called a step-up trans- 
former. If, however, the 
secondary contains the 
fewer turns, its potential 
difference is lower than that 
of the primary, and the transformer is a step-down trans- 
former. The ratio of the two potential differences is 
equal to the ratio of the number of turns of wire in the two 
coils. For example, a transformer that is to be used to 
reduce a voltage of 2000 to a lower one of 100 volts would 
be constructed by winding the primary coil with 20 times 
as many turns as the secondary. The same transformer 
would reduce a potential difference of 2200 volts to one 
of 110 volts. 



primary 



Secoadcny\ 




Fig. 382. — A Commercial Trans- 
former. 



446 A HIGH SCHOOL COURSE IN PHYSICS 

462. Utility of the Transformer. — The value of the 
transforming process is made clear by the study of a 
specific case. For example, electric power is to be trans- 
ferred from a power station to a large city over several 
miles of wire. It is desired that the number of amperes 
((7 in Eq. 8, § 437) be small in order to reduce to a mini- 
mum the loss of energy due to the heating of the con- 
ducting wires. The alternator used at the power house 
develops a potential difference of 2000 volts. This cur- 
rent is led through the primary coil of a " step-up " 

A B c 




2000 
Volts 
Alternator 





Fig. 383. — The Transformer System of a Long Distance Power Circuit. 

transformer A^ Fig. 383, where the potential difference 
is raised to 11,000 volts, while the number of amperes 
becomes proportionately reduced. At this voltage the 
current would be about 7 amperes per 100 horse power. 
Since wires of so great a potential difference are unsafe 
to lead into houses, the voltage is reduced from 11,000 to 
2000 volts, by a '' step-down " transformer B where the 
line enters the city, and again at the houses, from 2000 to 
100 volts by the transformer O. 

Since the power transmitted by an electric current is the product 
of the number of volts and amperes (Eq. 7, §436), a current of 10 
amperes under a potential difference of 11,000 volts, for example, 
delivers a power of 110,000 watts, which is about 147.5 horse power. 
It is plain, therefore, that a large amount of power under a high 
voltage can be transferred by a small current. The power generated 
at Niagara Falls is distributed to Buffalo, Syracuse, and other places 
with potential differences of from 22,000 to 60,000 volts. 

Again, if the transformer C, Fig. 383, deliver a current of 5 amperes 
under a potential difference of 100 volts for lighting a building, the 



ELECTRO-MAGNETIC INDUCTION 



447 



power delivered is 5 x 100^ or 500 watts. In the long distance circuit 
between A and B where the voltage is 11,000, the current would be 
500 4- 11,000, or 0.045 ampere. Hence, to take 5 amperes at 100 volts 
would increase the current in the long line only 0.045 ampere. 

The heat losses in long distance power transmission render it 
impracticable to convey large currents, as shown in § 438. But the 
examples above show that by raising the potential difference to sev- 
eral thousand volts the current is proportionately reduced ; conse- 
quently a given power can be transferred with far less heat loss, 
since the heat generated is proportional to the square of the current. 
It is in this manner that the transformer has solved the problem of 
long distance transmission of power from places where power is com- 
paratively cheap to distant manufacturing centers where it would be 
expensive. The distribution of power over long electric railway lines 
is accomplished by means of so-called high tension, i.e. high potential, 
apparatus. 

The alternating current transformer affords one of the 
most striking examples of the transformation and trans- 
ference of energy to be found. The power in one circuit 
is transferred to another entirely without any mechanical 
connection between the two. It is necessary only that the 
magnetic lines set up by the current in the primary coil 
pass through the secondary. No mechanical motion is 
concerned in the process. The power loss in a good 
transformer is usually not more than 3 or 4 per cent. 

463. Incandescent Lighting. — It is a 
familiar fact that the heating eff'ect of an 
electric current is employed in the process 
of electric lighting. The simplest case 
to study is the incandescent lamp. See 
Fig. 384. In this lamp the current is 
sent through a carbon filament 0^ which 
IS heated to incandescence. In order to 
prevent the carbon from burning, as well Fig. 384. — An incan- 

i ,1 £ -L. 1. X. j_- descent Lamp with 

as to prevent loss of heat by convection, carbon Filament 
it is inclosed in a highly exhausted glass and Socket. 




I 



448 A HIGH SCHOOL COURSE IN PHYSICS 

bulb. Connections are made with the ends of the filament 
by means of two short pieces of platinum wire sealed in 
the glass. One of these leads to the contact A in the 
center of the base, the other to the brass rim B wliich holds 
the lamp in its socket S. Through these the current is 
transmitted to and from the lamp. 

On account of its large consumption of power (about 
3.5 watts per candle power), many efforts have been made 
to produce lamps of higher efficiency. At the present 
time the carbon filament lamp is being rapidly replaced 
by those provided with metallic filaments of the rather 
uncommon metals tantalum or tungsten. These metals 
admit of being drawn out into very thin wires which can 
be heated white-hot without melting. These thin fila- 
ments are mounted in glass bulbs in much the same man- 
ner as the carbon filaments. Not only is the light which 
metallic filament lamps emit whiter than that given off by 
the carbon filaments, but the efficiency is far greater. In 
the tungsten lamp the consumption is about 1.25 watts per 
candle power. 

Ordinarily the potential difference on a lamp circuit is 
maintained at 110 or 220 volts. Lamps are adapted to the 
voltage of the circuit on which they are to be used. A 
16-candle-power lamp with a carbon filament requires a 
current of slightly more than 0.5 ampere when the voltage 
is 110 and about 0.25 ampere when the voltage is 220. 
The power necessary is, by equation (7), § 436, 110 x 0.5, 
or 55 watts. 

The Nernst lamp employs a short rod, or " glower," Fig. 385, com- 
posed of oxides which are maintained at a high temperature by the 
passage of a current of electricity. Although the glower is a non- 
conductor at ordinary temperatures, it becomes a conductor when 
heated. The glower is mounted close to a heater coil of fine platinum 
wire through which a current passes when the lamp is turned on. 



ELECTRO-MAGNETIC INDUCTION 449 

As soon as the glower becomes sufficiently hot, the heater coils are 
automatically thrown out of circuit, and the current then flows only 
through the glower. Since the glower is in- 
combustible, it can be used without being 
inclosed. The efficiency of the Nernst lamp 
is considerably below 2 watts per candle 
power. 

464. Incandescent Lamp Circuits. — 

Incandescent lamps are connected in 
parallel between the two main wires Fig. 385. — Showing Parts 
leading into a building. These wires ^^ ^ ^^'"'*^ ^^"^p- 
are maintained by the dynamo i>, Fig. 386, at a potential 
difference of 110 volts, so that any lamp may be turned 

on or off without interfer- 





ing with the others. Each 
16-candle-power lamp re- 
quires a current of half an 
ampere and, consequently, 
has a resistance when hot 

Fig. 386. — Showing the Connection of of 220 ohms. Lamps are 
Incandescent Lamps in a Circuit. ^f^en joined in grOUps, as 

shown at A. In this case the switch placed at S controls 
all the lamps of the group. 

465. Cost of Electric Power. — Electrical energy is sold 
at a certain rate per watt-hour. A watt-hour is a volt- 
ampere-hour ; in other words, an ampere of current flow- 
ing under a potential difference of one volt for one hour 
delivers a watt-hour of energy. Hence a 110-volt lamp 
carrying 0.5 ampere requires 110 x 0.5 x 1, or 55 watt- 
hours for every hour it is used. In commercial lighting 
a meter which is designed to register the consumption of 
energy in kilowatt- hours is placed in the circuit at the 
point w^here the wires enter each consumer's house. Thus 
readings of the meter show the amount of electrical energy 

for which the user is charged. 
30 



450 



A HIGH SCHOOL COURSE IN PHYSICS 



466. The Electric Arc. — The first electric arc was ex- 
hibited in 1809 by the great English scientist, Sir Hum- 
phry Davy. For this purpose Davy employed over 2000 
voltaic cells joined to two pieces of charcoal which were 
touched and then slightly separated. The same experiment 
may be easily made on any commercial lighting circuit. 

Wind bare copper wire, about No. 18, around pieces of electric 
light carbons. Join these in series with a resistance of about 10 ohms 
of iron wire to the terminals of a 110-volt lighting circuit. The re- 
sistance may be made by winding 150 to 200 feet of No. 18 or 19 iron 
wire on a suitable frame. Touch the tips of the carbons together and 
at once separate them about a quarter of an inch. An intensely 
bright light will be produced as the current continues to flow across 
the gap. 

By the separation of the carbon rods, a high tempera- 
ture is produced by the current, which vaporizes some of 

the carbon, forming a conducting 
layer from one to the other. The 
resistance of this mass of vapor may 
not be more than 3 or 4 ohms. If 
the current is a direct one, the tem- 
perature of the positive carbon rises 
above that of the negative, and from 
it comes the greater portion of the 
light. In this case the positive car- 
bon is consumed about twice as fast 
as the negative and becomes hollowed 
out, as in Fig. 387, while the negative 
remains pointed. When an arc is 
produced by an alternating current, 
light is given out equally from the 
two points, and the two rods are con- 
sumed at the same rate. 

467. Arc Lamps. — With the development of the dy- 
namo, the arc lamp as a means of illumination has come 




Fig. 387.— The Electric 
Arc Produced by a Di- 
rect Current. 



ELECTRO-MAGNETIC INDUCTION 



451 





The Hand-feed Electric 
Lamp. 



into extensive use. In the so-called liand-feed lamp, 

Fig. 388, the positions of the carbons are controlled by 

the screw heads aS'. They are 

at first permitted to touch 

and then are separated. As 
fast as the 
rods are 
consumed, 
they are 
slowly fed 
together Fig. 388. 
by the op- 
erator. Such lamps are used mainly in 
projection lanterns (§ 324). In the arc 
lamp of automatic feed. Fig. 389, the 
mechanism has two duties to perform: 
(1) to separate the carbons when the 
current starts, and (2) to 
feed the carbons together 
as fast as they are con- 
sumed, always keeping the 

„ „ _ proper space between 

Fig. 389.— The Auto- , 
matic Arc Lamp. tnem. 

The consumption of the carbon rods in the arc lamp 
can be largely reduced by inclosing the arc in a globe 
that is nearly air-tight as shown in Fig. 390. In this 
form of lamp the carbon burns from 60 to 100 hours. 

In the lamps just described the light is emitted by 
the incandescent ends of the carbons. If the carbons, 
however, are cored with a mixture of carbon and 
metallic salts, a highly luminous vapor is maintained 
by the current between the two terminals. In flaming 
arc lamps the carbons are cored with calcium salts which serve to give 
the light a bright yellow color. 

The open arc is operated by a current of from 5 to 10 amperes and 
45 to 50 volts, and the candle power in the direction of greatest in- 




FiG. 390. — In- 
closed Elec- 
tric Arc 
Lamp. 



452 A HIGH SCHOOL COURSE IN PHYSICS 

tensity is about 1000. The inclosed arc lamp requires a voltage of 
about 80 and a current of from 5 to 8 amperes. 

EXERCISES 

1. According to Lenz's law, in what direction in a circuit will a 
current be induced by the sudden interruption of a current in a neigh- 
boring conductor ? 

2. If the experiment of § 450 be repeated with a long loop of wire, 
it will be found that no deflection of the galvanometer will be produced 
by swinging both parts of the loops together across the earth's lines of 
force. Explain. By swinging the two parts of the loop in opposite 
directions, a large deflection is obtained. Why? 

3. Would you class the induction coil as a "step-up" or a "step- 
down " transformer ? 

4. Explain why the interrupter is an accessory part of the induc- 
tion coil but not of the transformer. 

5. A transformer carries a current of 10 amperes in its primary 
coil, under a potential difference at its terminals of 2000 volts. If it 
delivers a current of 194 amperes with a potential difference of 100 
volts, how much energy is transformed per hour, how much delivered, 
and how much wasted? 

6. If a building contained 50 110-volt incandescent lamps, what 
voltage w^ould have to be supplied to operate them if they were all 
joined in series? Show that this would be impracticable. 

7. Show by a diagram the manner of connecting ten 16-candle- 
power 110-volt incandescent lamps in parallel. What would have to 
be the voltage and how much current would be required ? 

Suggestion. — Consider that in the case of parallel conductors the 
total current is the sum of that in the separate parts. 

8. Why would the alternating-current transformer be entirely in- 
eifective in transforming a continuous current? 

9. Compute the monthly cost of operating five 16-candle-power in- 
candescent lamps when current costs 10 ct. per kilowatt-hour, allow- 
ing an average use of 3 hr. per day and 3| watts per candle power. 

10. Compute the monthly cost of current for an open arc lamp re- 
quiring 8 amperes at 50 volts, allowing 3 hr. per day and 10 ct. per 
kilowatt-hour. 

11. If the potential difference between the trolley of an electric 
railway and the track is 550 volts, show how it would be possible to 



ELECTRO-MAGNETIC INDUCTION 



453 



operate 110-volt incandescent lamps by properly connecting them 
together. Diagram the system. 

12. Show how a workman standing on the top of an electric car 
can safely handle the trolley wire with bare hands, while one standing 
on the ground would be severely shocked by coming in contact with 
any wire forming a connection with the trolley wire. 

13. 25 street lamps each operated by a potential difference at its 
terminals of 45 volts are joined in series. AVhat potential difference 
must be maintained at the terminals of the dynamo? What would 
have to be the current strength? Disregard the line loss. 

14. What has been the effect of economical long distance power 
transmission upon manufacturing industries, railway development, 
etc. ? 



4. THE TELEGRAPH AND THE TELEPHONE 

468. The Morse Telegraph. — The most extensive use 
of the magnetic effect of electric currents is made in the 
telegraph systems in common use. In 1831 Joseph Henry 
produced audible signals at a distance, but the system 
generally employed in this country is that designed by 
Samuel F. B. Morse and first used in 1844. The instru- 
ments found at each station are the key, sounder, and re- 
lay. 

The hey, Fig. 391, is 
merely a convenient 
device for making and 
breaking an electric cir- 
cuit at A by operating 
the lever L. It is also 
provided with a switch 
S^ so that the circuit 
may be left closed when 
the key is not in use. 

The sounder^ Fig. 
392, consists of an elec- 
tro-magnet M which at- fig. 392. — Telegraph Sounder. 




454 



A HIGH SCHOOL COURSE IN PHYSICS 



tracts the iron armature A whenever a current is sent 
through it, thus causing tlie heavy brass bar B to be 
drawn down against O with a sharp click. When the cur- 
rent IS interrupted, the armature is no longer attracted, 
and the bar is lifted against D by an adjustable spring. 
Transmitted messages are read by ear from the clicks of 
this instrument. 

469. Plan of a Short Telegraph Line. — Short lines in 
which the resistance is small require at each station only 
the key and sounder, as in Fig. 393. The connection is 




Earth 



Earth 



Fig. 393. — A Short Telegraph Line. 



usually made by a single wire, the circuit being completed 
by joining a wire at each end to a metal pipe or to a metal 
plate buried in the ground. Thus the earth forms a part 
of the circuit. The battery may be placed anywhere in 
the line. 

When the operator at Station A wishes to send a mes- 
sage to Station B^ he opens the switch on his key, which 
breaks the circuit. The sounder cores are thus demag- 
netized, and the bars are thrown up by the springs. Now, 
by means of the key, the operator A can make and break 
the circuit which will cause both sounders to click off the 
" dots " and " dashes " composing the message. A dot is 
produced by a quick stroke of the key which closes the 
circuit for only an instant ; a dash is a slower stroke 
which leaves the circuit closed for a slightly longer time. 



ELECTRO-MAGNETIC INDUCTION 



455 



The operator at B^ skilled in the interpretation of the 
clicks of the sounder, reads and records the message. 
Since a telegraph circuit is left closed when not in use, a 
" closed circuit," cell like the Daniell or gravity must be 
used. 

The Morse code given below is composed entirely of 
dots, dashes, and spaces. A small space is left between 
letters and a slightly longer one between words. 





The Morse 


T 


ELEGRAPH CODE 




a 


h 




O - - 


n 


b 


i - - 




P 


V 


c - - - 


] 




q 


w 


d 


k 




r - - - 


X 


e - 


1 




s 


y 


f 


m 




t — 


z 


g 


11 









local 
, Circuit 




The Telegraph Relay. 



470. The Relay and Its Use. — In long telegraph lines 
on which there are many instruments, the resistance is 
usually so great that 
the current in the main 
line is too feeble to op- 
erate the sounders with 
sufficient loudness. The 
difficulty is avoided by 
the use of the relay^ 
Avhich is a more sensitive instrument than the sounder. 
The relay, Fig. 394, consists of an electro-magnet M con- 
taining several thousand turns of wire (about 150 ohms) 
which is placed in the main line at each station together 
with the key. The armature and bar H of the relay are 
made very light, and all the adjustments of the instru- 
ment may be made with great precision. The function 
of the relay is to open and close at ^ a local circuit which 
contains merely the sounder and two or three cells. It 



456 



A HIGH SCHOOL COURSE IN PHYSICS 



will readily be seen that since there is little resistance to 
reduce the current, very distinct clicks will be produced 
on the sounder by the current from this local battery every 
time the relay automatically closes and opens the local 
circuit. 

471. The Long Distance Telegraph System. — The usual 
arrangement of the parts of a telegraph system is shown 
diagrammatically in Fig. 395. In sending a message from 




A Long Distance Telegraph System. 



Detroit to Buffalo, for example, the Detroit operator uses 
precisely the method described in § 469. When the circuit 
is opened at Detroit, no current flows from the main line 
battery, and all the relays on the line release their arma- 
tures and thus open every local circuit. When the De- 
troit operator presses his key, the main line battery sends 
a current out over the line, and the electro-magnets of the 
relays draw the armatures down and thus close the local 
circuits, causing every sounder to produce a sharp click. 
Thus the dots and dashes comprising a message may be 
read by every operator along the line. 

The relay may also be used to repeat a message to 



ELECTRO-MAGNETIC INDUCTION 



457 



another line instead of transmitting it to a short local 
circuit. A message from Chicago to New York may be 
repeated to a Detroit-Buffalo line at Detroit, to a Buffalo- 
Syracuse line at Buffalo, and finally to a Syracuse-New 
York line at Syracuse. Since the repeating is performed 
automatically at each of these stations, no time is lost in 
the transfer from one line to another. The relay when 
used in this manner is called a repeater. 

472. The Telephone. — The simplest manner in which 
speech produced at one station can be reproduced by elec- 
trical means at another is by means of two telephone 



Line 





Fig. 396. — A Simple Telephone System. 

" receivers " connected by two wires, or by one wire and 
the earth. See Fig. 396. The telephone receiver was 
invented in 1876 almost simultaneously by Alexander 
Graham Bell and Elisha 
Gray, both Americans. It 
consists simply of a perma- 
nent bar magnet Ji", Fig. 397, 
surrounded at the end by a 
coil of fine insulated copper 
wire C. An iron disk D is 
mounted so as to vibrate freely close to the end of the 
magnet. 




Fig. 397. 



Section of a Telephone 
Receiver. 



458 



A HIGH SCHOOL COURSE IN PHYSICS 




If a person speaks into the receiver, the sound waves 
set the disk in vibration. Each vibration of the disk 
changes the number of magnetic lines of force through 
the coils of vv^ire and thus induces a current whose nature 
depends entirely upon the loudness, pitch, and quality of 
the sound. In this manner a pulsating current is sent 
over the line to a similar instrument at the distant station. 

When the current generated at the 
first station flows in such a direction 
as to strengthen the magnet at the 
second one, the disk is drawn in; 
when it flows in the opposite direc- 

FiG. 398. — A Recent Type tion, the magnet is weakened and the 
of Receiver. ^ -i • 

disk released. Ihus the vibrations 
at the first station are reproduced on the instrument at the 
second. A modern receiver is shown in Fig. 398. 

473. The Transmitter. — The telephone receiver just 
described is not sufficiently powerful when used as a trans- 
mitter^ or sender, of speech ; but it is a receiver^ or repro- 
ducer, of sound of extremely great sensibility. For this 
reason the transmitters in general use 
are based on an entirely different 
principle from that of the receiver. 
The modern form used in long-dis- 
tance telephony consists of two car- 
bon buttons c and c'. Fig. 399, 
between which are carbon granules g. 
The metal disk, or diaphragm, D is 
attached to the button c. When the 
disk is set in vibration by the sound waves, the variation 
of pressure against the carbon granules causes large vari- 
ations in the electrical resistance between the buttons. 
Hence, if such an instrument be placed in a battery circuit 
and so connected that the current passes through the 




Fig. 399. — The Trans- 
mitter. 



ELECTRO-MAGNETIC INDUCTION 



459 



granules from c to c\ the strength of the current changes 
precisely in accordance with the vibrations of the diaphragm. 
474. A Long Distance Telephone System. — In all tele- 
phones operating with a local battery, the connections are 
made as shown in Fig. 400. Tlie current from the local 



Line Wire 



p Induct 



Induction 



E Co/7 

^_ ^Transmitter 




Earth 



Earth 




Fig. 400. — A Long Distance Telephone System. 

battery B is led through the transmitter and the primary 
of a small induction coil back to the battery. The main 
line contains at each station the secondary of the induction 
coil and the receiver. Two wires are generally used to 
connect the two stations, although one may be replaced by 
the earth. 

When a person speaks into the transmitter, the vibration 
of the diaphragm changes the pressure at the contact 
points of the carbon granules which conduct the current 
flowing through the primary coil. When the diaphragm 
is forced in, the resistance of the transmitter is lowered, 
and a comparatively large current flows ; when it moves 
outward, the current is reduced. These changes in the 
primary coil induce currents in the secondary which pass out 
over the line and set up vibrations in the receiver at the dis- 
tant station. 

For tlie purpose of calling attention, an electric bell is 
placed at each station. When the receiver is lifted from 
the hook upon which it hangs, the bells are disconnected 
from the line, and the connections are made as shown in 



460 A HIGH SCHOOL COURSE IN PHYSICS 

Fig. 400. The downward motion of tlie hook restores the 
bell to the line when the receiver is hung up. 

In cities and villages the telephones are all connected 
with a central exchange, where the operator upon request 
connects the line from any instrument with the line lead- 
ing to any other. Such exchanges can now be found in 
even the small towns and will serve to show to the student 
of electricity one of the ways in which the study of physi- 
cal principles has contributed to the prosperity and con- 
venience of mankind. 

EXERCISES 

1. Compute the resistance of an iron telegraph line 150 mi. long, 
the size of the wire being 0.1 in., the line containing also 10 relays of 
150 ohms each. Allow nothing for the earth connections. 

2. A telegraph wire offers a resistance of 35 ohms per mile. If the 
line contains five 150-ohm instruments, what current will be produced 
by 30 Daniell cells of 2 ohms each in a line 80 mi. long? Do you 
think this current would operate a sounder ? 

3. Connect a telephone receiver with the terminals of a very sensi- 
tive galvanometer and press in on the diaphragm. A deflection will 
be produced. Why? Release the diaphragm. A contrary current 
will be obtained. Why? 

4. If the telephone receiver can be used as a transmitter and re- 
quires no battery in its operation, why is it not so used? 



SUMMARY 

1. Currents of electricity maybe produced by induction 
by increasing or decreasing the number of magnetic lines 
of force which thread through a coil of wire if it is in a 
" closed circuit." The induced E. M. F. is proportional to 
the rate of change in the number of lines (§§ 446 and 447). 

2. The direction of an induced current is always such 
as to produce a field that tends to prevent a change in the 
number of lines of force through the coil (§ 448). 



ELECTRO-MAGNETIC INDUCTION 461 

3. When an electrical conductor cuts magnetic lines of 
force, an E. M. F. is induced within it (§ 450). 

4. Currents of electricity are produced on a large scale 
by making use of the dynamo^ which depends for its action 
on the principles of electro-magnetic induction. Dynamos 
are alternators or direct- current dynamos according as they 
produce alternating or direct currents (§§ 451 to 458). 

5. Dynamos are series-wound^ shunt-wound^ or compound- 
wound^ depending on the manner in which the field cores 
are excited (§ 457). 

6. The direct-current motor depends upon the tendency 
of a conductor carrying an electric current to move in a 
magnetic field. Motors are used wherever electric power 
is to be converted into mechanical power for moving 
machinery (§ 459). 

7. Alternating-current transformers are used to convert 
alternating currents of low potential into alternating cur- 
rents of high potential and vice versa. By their use 
unsafe currents of many thousand volts can be reduced to 
safe ones for domestic and commercial use (§§ 461 to 464). 

8. Electric power is sold by the kilowatt-hour. The 
number of kilowatt-hours consumed is given by the equa- 
tion 

kilowatt- hours = 0.001 EO x hours (§ 465). 

9o The telegraph and telephone employ the magnetic 
effect of the electric current in transmitting messages 
(§§ 468 to 474). 



CHAPTER XXI 
RADIATIONS 

1. ELECTRO-MAGNETIC WAVES 

475. An Electrical Discharge is Oscillatory. — When an 
electric spark jumps across a short gap, it appears to be 
only a single flash. The eye is incapable of determining 
whether or not this is actually the case. The following 
experiment may be used to investigate the nature of such 
a discharge. 

Bend 2 or 3 feet of wire into a hoop R, Fig. 401, leaving a gap of 
about 1 milhmeter at S. Connect the hoop in series with a Leyden 
jar L and the spark gap of an induction coil 
/ as shown. If the induction coil be now put 
into operation, sparks will be observed to pass 
across S every time they jump across the gap 
at I, 




The spark at S indicates that the air 
gap offers a better path than the metal 
loop STR. But we know that the re- 
sistance of the wire STR is but a frac- 
tion of an ohm, while that of the air 
at S is perhaps millions of ohms. The 
discharge through the loop, therefore, 
must meet with some impedance other 
than that which would be encountered by a direct current. 
From this and other effects we are led to infer that the 
discharge at I is a rapid surging of electricity hack and 
forth^ but one which lasts only for a fraction of a second. 
At the first rush of current in the ring a magnetic field is 

462 



Fig. 401. — Illustrat- 
ing an Effect of 
an Electric Dis- 
charge. 




RADIATIONS 463 

suddenly set up within the loop. This sudden magnetic 
change induces, according to Lenz's law (§ 448), an op- 
posing E. M. F. which effec- 
tually prevents the flow of a 
large portion of the current 
in the loop. Likewise the 

sudden reversal of the dis- Fig. 402.— Result Obtained by Photo- 
charge reverses the magnetic graphing an Electric Spark. 

lines and again inducfes an opposing E. M. F. in the loop. 
Hence the greater portion of the discharge finds a better 

outlet through the gap S. 
Stronger proof of the oscillat- 
ing nature of a discharge has 
been obtained by photograph- 
ing a spark by the help of a 
revolving mirror. The result 
is shown in Fig. 402. The 
Fig. 403. — Diagram of a Spark period of oscillation lias been 
^^^^' shown by this method to be 

of the order of a millionth of a second. As a rule the 
oscillations subside very quickly, as shown by the curve 
in Fig. 403. 

476. Electro-magnetic Waves in the Ether. — In 1888 
Hertz 1 of Germany showed that each electrical oscillation 
that occurs when a spark passes across an air gap pro- 
duces a disturbance in the surrounding ether which is 
propagated outward in wave form in much the same 
manner as water waves move outward from a falling 
pebble, or sound waves from a vibrating bell. These 
ether waves are called electro-magnetic waves. Inasmuch 
as light itself is propagated in wave form in the ether, 
it might be inferred that the speed of the two should be 
the same. Such has been found to be the case. 

1 See portrait facing page 464. See also Kelvin, frontispiece. 





464 A HIGH SCHOOL COURSE IN PHYSICS 

477. Detection of Electro-magnetic Waves. — The process 
of transmitting messages by means of electro-magnetic 
ether waves is dependent on the detection of such waves 
at the receiving station. One method that can be em- 
ployed for this purpose is illustrated by the following 
experiment. 

Bend a glass tube about 5 centimeters long and 3 or 4 millimeters 
in diameter as shown in Fig. 404. Place a few coarse iron filings in 
the bend and introduce a globule of clean mercury into each end of 
the tube as shown at A and B. Insert small 
iron wires into the mercury and connect the 
device in series with a cell and an ammeter or 
galvanometer. No deflection should be pro- 

-pjQ 4Q4 ^ Cuherer ^^^^ced. Now cause a spark to jump a short air 

gap several feet away by using an induction 
coil, an influence machine (§ 371), or a Ley den jar. A deflection 
will be observed at once. If the tube of filings be now tapped lightly, 
the circuit is again broken, and the experiment may be repeated. 

The tube of metal filings used as a detector of electro- 
magnetic waves is called a coherer. Ordinarily the filings 
offer a large resistance to the flow of electricity through 
the many points of contact between the several pieces. 
The waves emitted by the oscillatory discharge at the 
spark gap cause the filings to cling together, and the 
resistance of the tube immediately falls to a few ohms. 
A tap or jar breaks the filings apart, and the resistance 
rises again to its former value. 

478. Wireless Telegraphy. — The possibility of sending 
out electro-magnetic waves from an induction coil and of 
detecting them by a coherer has led to the transmission 
of messages by the so-called wireless telegraph systems, 
A plan of a simple sending and receiving equipment is 
shown in Figs. 405 and 406. 

Connect the coherer described in the preceding section in series 
with the electro-magnet of a relay R and a cell C, Fig. 406. In the 




HEINRICH HERTZ (1857-1894) 



About the middle of the last century Maxwell advanced the idea 
that waves of light are electro-magnetic in character. If this were 
true of light, then it would also be true of radiant heat. In 1888 
Hertz of Germany succeeded in demonstrating experimentally the 
truth of this assumption. During the discharge of electricity between 
two polished knobs, so-called electro-magnetic waves are radiated into 
space. Hertz was able to detect these waves and to reflect, refract, 
and polarize them and to cause interference to take place between 
them. He also measured the velocity with which they are propa- 
gated through space, and found it to be equal to the velocity of 
light. Thus the hypothesis of Maxwell was placed upon an experi- 
mental basis, and the way opened for long-distance communication 
between stations without the necessity of connecting wires. The 
practical value of Hertz's results can hardly be overestimated. 
Among others who have contributed greatly to the knowledge of 
electro-magnetic waves may be mentioned Sir Oliver Lodge of Eng- 
land, Righi of Bologna, and Branly of Paris. 

In 1880 Hertz was made assistant to Helmholtz at Berlin. In 
1885 he became professor of physics in the Technical High School 
at Karlsruhe. It was in the latter place that his epoch-making 
experiments were first performed. In 1889 he was elected professor 
of physics at Bonn, where he died at the age of thirty-seven years. 
Electro-magnetic waves are called Hertzian waves in his honor. 



RADIATIONS 



465 



make-and-break circuit of the relay connect an electric bell B (§ 411) 
and a cell C. The bell is so placed that its hammer strikes the co- 
herer D whenever it is set in vibration. One end of the coherer 
should be connected to the earth and the other joined to a high aerial 
wire. For short distances the aerial wire may be very short or left 
off altogether. At the sending station (Fig. 405), simply connect 
one side of a spark gap of an induction coil with the earth and 
join the other side with an aerial conductor which is merely a 
rod or wire, one end of which is lifted some distance above 
the apparatus. A key K should be placed in the circuit of 
the primary coil. Closing the key and thus producing sparks 
at S causes the bell to ring at the receiving station until the 
circuit at the sending station is broken. 



•s 



J 



nffi-i^ 



U 



Ih 



When sparks pass at the so- 
called oscillator /S", electro-mag- 
netic waves are emitted which 
affect the coherer at the receiving 
station, causing the filings to con- 
duct as shown in S 477. The cur- 



'" '~Earth 

Fig. 405. — Appai-atus for 
Sending Wireless Tele- 
graph Signals. 

rent from the cell O then flows 
through the coherer and the 
electro-magnet of the relay 
R. The relay closes at A the 
circuit through the bell which 
is rung by" the current from 
the cell C . The filings con- 
tinue to cohere until sparks 
cease to pass at S^ when the 
taps of the bell hammer jar 




them apart. 
31 



The circuit 



in'rth 

Fig. 406. — A Receiving Station for 
Wireless Messages. 



466 



A HIGH SCHOOL COURSE IN PHYSICS 



through the relay is thus broken, which in turn opens 
the bell circuit at A. Another spark at ^S' again causes 
D to conduct and the bell to ring. Many other forms 
of receiving devices are in common use. 

Systems of wireless telegraphy have reached such a 
state of development that ocean-going vessels are, as a 
rule, at all times in communication with land stations or 
with one another. Passengers on sinking vessels have 
thus been rescued by timely assistance obtained through 
wireless messages which were received at stations many 
miles away. Naval fleets equipped with a good system 
may be kept continually informed by the controlling 
department of the government, and the department may 
be kept acquainted with every movement of the fleet. 
Even transatlantic messages are now transmitted without 
the use of wires or cables. 



2. CONDUCTION OF GASES 

479. Conduction through Vacuum Tubes. — Connect the ter- 
minals of an induction coil with electrodes A and />, Fig. 407, which 

are sealed in the ends of a glass 
tube 2 or 3 feet in length. Con- 
nect the tube with an air pump 
and put the coil in operation. 
Sparks will jump across the gap S 
because it is the better path. 
Put the air pump in action and 
exhaust the tube while the coil 
is running. When the pressure 
has been sufficiently reduced, the 
discharge will begin to take place 
through the long exhausted tube 
rather than over the shorter path through the air at S. 

When the exhaustion of the gas has been carried much 
farther, there may be seen a radiation from the cathode 
proceeding in straight lines and traceable by a slight 




Fig. 407. — Discliarjje through a Par- 
tial VaQuum. 



SIR WILLIAM CROOKES (1832- ) 

The discharge of electricity through 
partially exhausted tubes has been a sub- 
ject of much research since the middle of 
the last century. In 1853 Masson of Paris 
discharged electricity from a large induc- 
tion coil through a Torricellian vacuum. 
Later Geissler, a German, constructed ex- 
cellent tubes containing small amounts of 
different gases, which became famous on 
account of the great beauty of color mani- 
fest upon discharging electricity through 
them. Crookes began his experiments in 
1873 with tubes in which the exhausti—i 
was carried to a high degree. In these he 
showed that so-called " radiant matter " is 
thrown out from the electrodes in straight lines, casts shadows when 
intercepted by solids, and is capable of producing mechanical effects 
when brought into collision with light movable vanes. Furthermore, 
he showed that the stream of particles is deflected from a straight 
line by a magnet. 




WILHELM KONRAD RONTGEN (1845- ) 



The most striking phenomena accom- 
panying discharges in highly exhausted 
tubes were discovered by Professor Ront- 
gen at Wurzburg, Germany, in 1895. A 
Crookes tube was left in operation on a 
table. Beneath it was a book containing 
a key as a bookmark; below the book was 
a photographic plate in a plate holder. 
Later, on using the plate in a camera, and 
developing it in the usual manner, a well- 
defined shadow of the key became visible 
upon it. Further experimentation showed 
the discoverer that he had found a new 
kind of radiation which he named X-rays. 
The important ends attained by the use of 
X-rays in the practice of medicine and surgery will always serve to 
keep the name of Rontgen before the public. 




RADIATIONS 467 

luminescence occasioned in the gas remaining in the tube. 
Where these rays fall upon the glass, it is made warm 
and luminous. Ordinary glass glows with a soft greenish 
yellow light ; and a solid, as marble, placed in the path 
of the radiations will glow with a characteristic color. 
These radiations are known as cathode rays. Such highly 
exhausted and hermetically sealed tubes are known as 
Crookes' ^ tubes, 

480. Cathode Rays. — Cathode rays are characterized 
by three main properties, viz. (1) they are deflected from 
a straight line when caused to pass through an electric or 
a magnetic field, (2) they convey a negative charge of 
electricity to the object upon which they fall, and (3) they 
raise the temperature of any solid object which obstructs 
their path. The inference is, therefore, that cathode rays 
are- swiftly moviyig imrtleles charged with negative electricity. 
The velocity with which the particles move sometimes 
reaches the enormous value of over 50,000 miles per 
second, — nearly one third of the velocity of light. A 
continuous stream of such particles, all of which carry 
negative charges, is equivalent to a current of electricity ; 
hence their deviation in passing through a magnetic field. 

481. X-Rays. — The most important application of the 
action of cathode rays is employed in the production of 
the so-called X-rays, 
which were discovered 
in 1895 by Rontgen,i a 
German physicist. A 

special vacuum tube of '^^^^K^ 

the form shown in Fio*. - — 

.^o . , „ ,. . " Fig. 408. — AnX-RayTube. 

408 IS used for this pur- 
pose. This is connected with the secondary of an induc- 
tion coil in such a manner that (7, a concave electrode, is 

1 See portrait facing page ^QQ, 




468 A HIGH SCHOOL COURSE IN PHYSICS 

the cathode, and P the anode. Since the cathode particles 
are sent off at right angles to the surface which they leave, 
they are focused against the solid piece of platinum P 
placed near the center of the tube. Accompanying the 
impact of the cathode rays against P appears a new radia- 
tion of an entirely different character. It is to these that 
the name of X-rays^ or Rontgen rays, has been given. 

482. Properties of X-Rays. — The properties which give 
to X-rays their great practical utility is their power (1) to 
penetrate bodies of matter that are opaque, and (2) to 
make a permanent impression upon photographic plates. 
Substances are not all equally transparent to the rays ; 
e.g. they more readily penetrate a thick book than a thin 
coin ; and while flesh is quite transparent, bones are more 
or less opaque. Hence, if the hand be held between an 
X-ray tube and a photographic plate, the bones will shield 
the plate more than the flesh and thus produce shadows. 
Fig. 409 shows an X-ray picture of a broken wrist, and 
the same wrist is shown in Fig. 410 after having healed. 

3. RADIO-ACTIVITY • 

483. Radio-active Substances. —Place a gas mantle that has 
been pressed out flat upon a photographic plate and inclose it in a 
light-proof box. After three or four weeks develop the plate in the 
usual way. A distinct image of the fabric of the mantle will become 
visible. Such a plate is shown in Fig. 411. 

Similar experiments may be made with compounds con- 
taining uranium, especially by using the mineral uraninite, 
or pitchblende. These experiments show that certain sub- 
stances spontaneously emit a kind of radiation capable of 
affecting a photographic plate. Such substances are said 
to be radio-active^ or to possess the property of radio-activ- 
ity. The radio-active element contained in a gas mantle 
is thorium. The element radium is remarkable for its 




Pig. 409. — X-Ray Picture Showing 
the Fractured Bones of a Wrist. 



Fig. 410. — X-Ray Picture of the 
Wrist Shown in Fig. 409, after it 
has been Healed. 




Fig. 411. — The Radio-active Effect of a Portion of a Gas Mantle on a 

Photographic Plate. 



RADIATIONS 469 

intense radio-activity. The discovery of this element by 
Monsieur and Madame Curie ^ of Paris, in 1898, resulted 
from the fad that its presence in microscopic quantities 
in tons of the mineral from which it was taken was de- 
tected by its exceptionally large radio-active effects. 

484. Radio- Activity. — The property of radio-activity 
was discovered by Henri Becquerel ^ of Paris in 1896. In 
honor of the discoverer the radiations emitted by radio- 
active substances are often called Becquerel rays. Ex- 
perimental researches have shown that Becquerel rays are 
of a complex nature. They consist (1) of negatively 
charged particles called beta rays, (2) of positively 
charged particles called alpha rays, and (3) of radia- 
tions resembling X-rays in nature, which are called 
gamma rays. 

Becquerel rays are detected not only by their power to 
affect a photographic plate as shown in § 483, but also by 
their power to discharge electrified bodies near which they 
are placed. This effect is due to the fact that the air 
surrounding a radio-active body is rendered a conductor 
of electricity by the radiations emitted. Furthermore, 
if minute crystals of zinc sulphide be placed in the 
immediate neighborhood of an extremel}^ small quantity 
of radium, they show intermittent flashes of light as they 
are bombarded by the alpha particles expelled by the 
radium. 

485. Electrons. — Beta rays, or the negatively charged 
particles emitted by a radio-active substance, are called 
negative electrons. Negative electrons are separated from 
alpha and gamma rays on passing through a strong electric 
or magnetic field on account of the difference in the kind of 
charge carried by them. The path of the negative elec- 
trons is curved in one direction by the field, while that of 

1 See portrait facing page -470. 



470 A HIGH SCHOOL COURSE IN PHYSICS 

the alpha rays is bent in the opposite direction. The 
gamma rays remain unchanged in direction. 

Knowledge of electrons began with the dissovery by Sir 
J. J. Thompson of England that the cathode rays pro- 
duced in a vacuum tube (§ 480) consist of swiftly moving 
particles charged with electricity. These particles have 
since been found to be identical with the negative elec- 
trons emitted by a radio-active substance. The speed 
attained by the electrons in a cathode tube is about 
62,000 miles per second, but the electrons emitted by 
radium move with speeds that reach as high as 165,000 
miles per second, which is about -^^ the velocity of 
light. 

The mass of an electron has been ascertained and found 
to be about -j^q q the mass of an atom of hydrogen. Dif- 
ferences in the electrical, magnetic, and other properties 
of matter are attributed to variations in the arrangements 
and movements of the electrons associated with the mole- 
cules. For example, in electrical conductors the electrons 
are less firmly attached to the molecules than they are in 
non-conductors, and are consequently moved readily by 
potential differences. This motion of electrons through 
a conductor constitutes an electric current. 

486. Alpha Particles. — The alpha rays emitted by a 
radio-active substance are of atomic size and have been 
found to be charged with positive electricity. Experi- 
ments have also shown that they are atoms of the well- 
known gas helium, and that their velocity may reach as high 
as -^Q the speed of light. The energy that is developed 
by a radio-active body is due mainly to the alpha particles 
which it hurls outward with this enormous rate of motion. 
It is estimated that the total amount of energy that can 
be given off by a gram of radium is about equal to that 
developed by the combustion of a ton of coal. 



ANTOINE HENRI BECQUEREL (1852-1908) 




A new epoch in the theory of matter was 
inaugurated by the discovery of radio- 
activity in 1896 by Becquerel of Paris. It 
had long been known that compounds con- 
taining the element uranium would produce 
an effect upon photographic plates in the 
dark. This singular action had been attrib- 
uted merely to their property of phosphor- 
escence. Becquerel proved that all com- 
pounds containing uranium act similarly on 
plates, even those which are not phosphor- 
escent. This phenomenon corresponds to a 
continuous emission of energy, for which 
the old view of matter did not account. 



MADAME CURIE (1867- ) 



Soon after the discovery of radio-activ- 
ity, investigations were made by Madame 
Curie of Paris and others in order to as- 
certain if radio-activity were not a general 
property of matter. Various compounds 
were tested; but the strange property 
appeared to be confined to substances 
containing uranium and another element, 
thorium. However, it was observed by 
Madame Curie that certain pitchblendes 
(minerals containing oxide of uranium and 
other well-known elements) were many 
times more radio-active than uranium. It 
therefore seemed probable that an unknown 
element of great radio-activity was pres- 
ent in the mineral. From this grew up the celebrated experiments of 
Monsieur and Madame Curie which led to the discovery of the re- 
markable element radium. Although the separation of radium from 
minerals is attended with enormous difficulties on account of the 
small quantity which they contain, enough of it has been secured for 
experimental research in the laboratories of the world to lead to the 
generalizations in regard to the constitution of matter contained in 
the concluding sections of this book. 




RADIATIONS 471 

487. Disintegration of Matter. — The emission of elec- 
trons and of positively charged particles which are compar- 
able in mass with atoms of hydrogen leads at once to the 
conclusion that a radio-active substance must be continu- 
ally experiencing a molecular change. Although a gram 
of radium develops hourly an amount of heat equal to 100 
calories, its transformation is so slow that nearly 13 centu- 
ries would elapse before one half its mass would suffer a 
change in its nature. Experiments have shown that it is 
along with this transformation that the element helium is 
produced. Thus the radio-active elements are not perma- 
nent, but by the emission of electrons and alpha particles 
they are being constantly transformed into elements of 
smaller atomic weights. Uranium^ for example, is sup- 
posed to be breaking up and forming for one of its products 
the element radium ; radium in turn forms so-called radium 
emanation^ and so on through a series. It is while these 
transformations are going on that the alpha, beta, and 
gamma rays are emitted. 

According to the disintegration theory of matter ad- 
vanced by Rutherford and Soddy, the atoms of radio-active 
substances are unstable systems which break up spontane- 
ously with explosive violence, expelling a small portion 
of the fractured atom with great velocity. The remaining 
portion of each atom forms a new system of a smaller 
atomic weight and possessing properties differing from 
those of the atoms of the parent substance. This new 
substance may be unstable also and undergo an atomic 
change similar to the preceding. The process may con- 
tinue by stages until at last a stable form is finally 
attained. 

488. The Domain and Future of Physics. — The study 
of Physics is primarily an investigation of environment to 
the end that the knowledge obtained shall be conducive to 



472 A HIGH SCHOOL COURSE IN PHYSICS 

the comfort of mankind, and lead to an increase of man's 
power in his range of action. If this were all, the pursuit 
of physical science would be amply justified, for the field 
is a broad one, and the results to which the study has led 
are overwhelming. It has opened the entire world to 
the traveler, it has brought the West within speaking 
distance of the East, it has overthrown the dangers and 
solitude of the sea, it has brought distant worlds within our 
range of vision and exposed the sources of disease, it re- 
veals the secret of aerial flight, and, now, at the present 
rate of advancement man may soon acquire that knowledge 
which the human mind has long sought to secure in answer 
to the all-important question, " What is matter ? " 



INDEX 



Absolute temperature, 222. 

Absolute units of energy, 52. 

Absolute units of force, 30, 31. 

Absorption, of heat, 253 ; of ra- 
diant energy, 253 ; selective, 
254; spectra, 327. 

Accelerated motion, 17. 

Acceleration, 16; due to gravity, 
70; centripetal, 45. 

Activity, 53. 

Adhesion, 130. 

Aeronauts, altitude reached by, 
153. 

Aeroplane, 43. 

Air, buoyancy of, 154; compres- 
sibility of, 147 ; density of, 
138; pressure of, 139, 155; 
variations in pressure of, 141. 

Air brake, 161. 

Alpha particles, 470. 

Alternating currents, 434. 

Alternator, 435. 

Altitude, effect of, on barometer, 
153; effect of, on boiling point, 
242. 

ASialgamation of zinc, 379. 

Ammeter, 404. 

Ampere, sketch and portrait of, 
facing 406. 

Amplitude, 74; effect of, on in- 
tensity of sound, 174. 

Aneroid barometer, 143. 

Anode, 396. 

Antinode, 194, 201. 

Arc, lamp, 451; light, 450; elec- 
tric, 450; enclosed, 451; flam- 
ing, 451 ; open, 451. 

Archimedes, 119; principle of, 
119. 



Armature, 392, 434; drum, 438; 
Gramme ring, 437. 

Artesian wells, 114. 

Athermanous substances, 255. 

Atmosphere, as unit of pressure, 
142; density of, 138, 152, 153; 
humidity of, 239 ; pressure of, 
139. 

Atoms, 471. 

Attraction, electrical, 335 ; mag- 
netic, 361; molecular, 130. 

Balance, 10; spring, 10. 

Balloon, 155. 

Barometer, 142; aneroid, 143; 

self-recording, 143; utility of, 

144. 
Battery, storage, 399. 
Beats, 190; law of, 191. 
Becquerel, sketch and portrait of, 

facing 470. 
Bell, diving, 162; electric, 392. 
Binocular vision, 317. 
Bodies, falling, 70 ; thrown, 72. 
Boiling, 241; laws of, 242. 
Boiling point, of liquids, 243; on 

thermometers, 213. 
Boyle's law, 149. 
Bright-line spectra, 328. 
Bunsen photometer, 276. 
Buoyancy, of air, 154; of liquids, 

119. 

Caisson, pneumatic, 162. 

Caloric, 257. 

Calorie, defined, 225; electric 

equivalent of, 415; mechanical 

equivalent of, 258. 
Camera, 312. 



473 



474 



A HIGH SCHOOL COURSE IN PHYSICS 



Candle power, defined, 276 ; of 
lights, 275. 

Capacity, electric, 348. 

Capillarity, 132; in soil, 137; in 
tubes, 133. 

Capstan, 95. 

Cathode, 396. 

Cathode rays, 467. 

Cell, chemical action in, 377; 
Daniell, 385; dichromate, 384; 
"dry," 387; gravity, 385; Le- 
clanche, 386; local action in, 
379 ; storage, 399 ; theory of, 
378; voltaic, 375. 

Cells, in parallel, 411; in series, 
410. 

Center of gravity, 65; of mass, 
65; of oscillation, 79; of per- 
cussion, 79. 

Centimeter-gram-second system, 
6. 

Centrifugal force, 45. 

Centripetal acceleration, 45. 

Centripetal force, 45, 46. 

Charges, electrical, 335; mixing, 
348. 

Charging, by contact, 338 ; by in- 
duction, 342. 

Charles, law of, 222. 

Chemical effects of currents, 395. 

Chord, major, 182. 

Circuit, divided, 420; electric, 
377. 

Coefficient of expansion, of gases, 
220; of liquids, 220; of solids, 
217. 

Coherer, 464. 

Cohesion, 130. 

Coil, induction, 429; magnetic 
action of, 389. 

Cold storage, 245. 

Color, 321; and wave length, 
322; by dispersion, 321; by in- 
terference, 323 ; of films, 332 ; 
of objects, 322; of pigments, 
324; of transparent bodies, 
323; complementary, 323. 



Commutator, 436. 

Compass, 368. 

Component forces, 36. 

Component motions, 21. 

Composition, of forces, 36 ; of 
motions, 21. 

Concave lens, 301, 303. 

Concave mirror, 284, 287. 

Condenser, 349. 

Conduction, of electricity, 338 ; 
of gases, 466; of heat, 248. 

Conductors, of heat, 248; charge 
on outside of, 343. 

Conjugate foci, of lenses, 304; 
of mirrors, 289. 

Conservation of energy, 59. 

Convection, 249. 

Convex lens, 301, 305. 

Convex mirror, 284, 285. 

Cooling, by expansion, 259 ; arti- 
ficial, 245. 

Couple, 41. 

Critical angle, 297. 

Crookes, sketch and portrait of, 
facing 466. 

Crookes' tubes, 467. 

Curie, Mme., sketch and portrait 
of, facing 470. 

Currents, electric, 375; chem- 
ical effect of, 395; heating 
effect of, 394; induced, 425; 
magnetic effect of, 388 ; meas- 
urement of, 402; mutual effect 
of, 393. 

Curvature, center of, 284, 301. 

Curvilinear motion, 44. 

Daniell cell, 385. 

Declination, 369. 

Density, 10, 124; of air, 138; 

of liquids, 127; of solids, 124; 

atmospheric, 153; electric, 344. 
Dew point, 240, 
Diamagnetic substances, 362. 
Diathermanous substances, 255. 
Diatonic scale, 181. 
Dichromate cell, 384. 



INDEX 



475 



Dielectric, 350. 

Dip, magnetic, 371. 

Discharge, electric, 351 ; oscil- 
latory, 462. 

Dispersion, 321. 

Distillation, 243. 

Diver, 162. 

Drum armature, 438. 

"Dry" cell, 387. 

Dynamo, 432 ; alternating-cur- 
rent, 434 ; compound-wound, 
440; rule of, 432; series- 
wound, 439 ; shunt-wound, 
439; unipolar, 440. 

Dyne, 30. 

Earth's magnetism, 368. 

Ebullition, 237, 241; laws of, 
241. 

Echoes, 177. 

Eclipses, 272. 

Efficiency, 99; of lamps, 448, 
449 ; of simple machines, 99. 

Elastic force in gases, 148. 

Electric bell, 392. 

Electric car, 443. 

Electric charge, vmit of, 348 ; dis- 
tribution of, 343. 

Electric motor, 442. 

Electrical attraction, 335. 

Electrical circuits, 377. 

Electrical currents, 375. 

Electrical machines, 354. 

Electrical repulsion, 336. 

Electrical resistance, 405. 

Electricity, current, 375; static, 
335. 

Electrolysis, 396. 

Electro-magnet. 391. 

Electro-magnetic induction, 425. 

Electromotive force, 376 ; in- 
duced, 426, 431. 

Electrons, 469. 

Electrophorus, 352. 

Electroplating, 397. 

Electroscope, 336. 

Electrotyping, 398. 



Energy, 54 ; of the sun, 255 ; 
conservation of, 59 ; electric, 
413; equation for, 56; heat, 
210; kinetic, 56; potential, 
55; radiant, 253; transfer- 
ences and transformations of, 
58. 

Engine, gas, 261; steam, 260; 
turbine, 263. 

English equivalents of metric 
units, 5, 6, 8, 53. 

Equilibrant, 38. 

Equilibrium, 66; neutral, 67; 
stable, 60 ; unstable, 66. 

Erg, 52. 

Ether, 252, 268, 463. 

Evaporation, 237; laws of, 237. 

Expansion, 216; of gases, 147 / _ _ 
221; coefficient of, 217; un- 
equal, 218. 

Extension, 2, 4. 

Eye, 312. 

Falling bodies, 69. 

Faraday, sketch and portrait of, 
facing 426. 

Field, electric, 340; magnetic, 
363. 

Field-magnet of dynamo, 439. 

Floating bodies, 122. 

Floating dry dock, 123. 

Fluids, 138. 

Focal length, of lens, 302; of 
mirror, 285. 

Focus, conjugate, 289, 304; prin- 
cipal, 284, 302; real, 285; vir- 
tual, 285, 304. 

Foot-pound, 52. 

Foot-pound-second system, 10. 

Force, 29 ; centrifugal, 45 ; cen- 
tripetal, 45, 46; component of, 
36; composition of, 36; field 
of, 340; lines of, 340, 303; 
moment of, 39 ; parallelogram 
of, 37 ; representation of, 35 ; 
resolution of, 42; units of, 
30, 31. 



476 



A HIGH SCHOOL COURSE IN PHYSICS 



Forces, parallel, 40. 

Franklin, sketch and portrait of, 

facing 346. 
Fraunhofer lines, 327. 
Freezing, 231; heat given out 

during, 235. 
Friction, 99; rolling, 100. 
Fundamental tone, 194, 200. 
Fusion, 231; of ice, 234; heat of, 

233; laws of, 232. 

Galileo, sketch and portrait of, 

facing 70. 
Galvanometer, 382 ; astatic, 403 ; 

d'Arsonval, 403; tangent, 403. 
Gas engine, 261. 
Gases, characteristics of, 138; 

compressibility of, 147. 
Gilbert, 368. 

Gram, of force, 32 ; of mass, 7. 
Gramme ring, 437. 
Gravitation, law of, 62. 
Gravity, acceleration due to, 70 ; 

center of, 65 ; force of, 63 ; 

variation of, 70. 
Gravity cell, 385. 
Guericke, Otto von, 155. 

Harmonic, 196. 

Heat, 210; due to absorption of 
radiant energy, 255 ; due to 
electric current, 415; of fusion, 
233; of sun, 255; of vaporiza- 
tion, 249; conduction of, 248; 
convection of, 249; measure- 
ment of, 225; mechanical' 
equivalent of, 258; produced 
by compression, 210; produced 
by friction, 210; radiation of, 
252; specific, 226; unit of, 
225. 

Heating, by hot air, 250; by hot 
water, 251; electrical, 394. 

Helmholtz, sketch and portrait 
of, facing 196. 

Henry, sketch and portrait of, 
facing 426. 



Hertz, sketch and portrait of, 
facing 464. 

Horse power, 53; electric equiva- 
lent of, 415. 

Humidity, 240. 

Huyghens, 79, 268. 

Hydraulic elevator, 117. 

Hydraulic press, 112. 

Hydraulic ram, 118. 

Hydrometers, 127. 

Hydrostatic paradox, 109. 

Ice, artificial, 245. 

Images, by double reflection, 282 ; 
by lenses, 305, 307; by plane 
mirror, 280 ; by spherical mir- 
rors, 284; pin-hole, 273; real, 
287; size of, 310; virtual, 287. 

Incandescent lamp, 447. 

Incandescent lighting, 447, 449. 

Incidence, angle of, 279, 293. 

Inclination of magnetic needle, 
371. 

Inclined plane, 96. 

Index of refraction, 294; meas- 
ured, 295. 

Induced charges, 342. 

Induced magnetism, 362. 

Induction, charging by, 342; 
electro-magnetic, 425; electro- 
static, 340; magnetic, 362. 

Induction coil, 429. 

Inertia, 29. 

Influence machine, 354. 

Insulators, 338. 

Intensity, of light, 275; of 
sound, 173. 

Interference, of light, 330; of 
sound, 188. 

Intervals, 182. 

Ions, 378, 396. 

Isobars, 145. 

Isogonic lines, 370. 

Jackscrew, 98. 
Jar, Leyden, 351. 



INDEX 



477 



Joule, sketch and portrait of, 

facing 258. 
Joule's equivalent, 258. 
Joule's law, 415. 

Kelvin, Lord, sketch and por- 
trait of, frontispiece. 

Key, tone, 181; telegraph, 453. 

Kilogram, standard, 7. 

Kilogram-meter, 52. 

Kinetic energy, 56; equation of, 
56. 

Kinetoscope, 317. 

Kirchoff, 328. 

Lamps, arc, 450; incandescent, 
447; Nernst, 448. 

Lantern, projection, 316. 

Laws of motion, 28. 

Leclanche cell, 386. 

Lenses, 300 ; achromatic, 322 ; 
concave, 301, 303; converging, 
302; convex, 301, 305; diverg- 
ing, 303; equation of, 310. 

Lenz's law, 427. 

Level, liquid, 110. 

Lever, 88; kinds of, 89; mechan- 
ical advantage of, 89. 

Leyden jar, 351. 

Light, dispersion of, 321; inter- 
ference of, 330; meaning of, 
268; reflection of, 278; refrac- 
tion of, 292; speed of, 269. 

Lightning, 345. 

Lightning rods, 345. 

Lines, of force, 340; isogenic, 
370. 

Liquefaction, see Fusion. 

Liquid, in communicating ves- 
sels, 110; density of, 127; pres- 
sures in, 103 ; pressures trans- 
mitted by, 111; surface ten- 
sion of, 132; thermal conduc- 
tivity of, 248. 

Liter, 6. 

Local action, 379. 

Lodestone, 359. 



Longitudinal loops, 194. 
Longitudinal waves, 171. 
Loudness of sound, 173. 

Machines, efficiency of, 99; elec- 
trical, 354; general law of, 84; 
simple, 83. 

Magdeburg hemispheres, 147. 

Magnet, artificial, 361; electro-, 
391; horseshoe, 361; natural, 
359; poles of, 360. 

Magnetic attraction, 361. 

Magnetic field, 363. 

Magnetic induction, 362. 

Magnetic lines of force, 363. 

Magnetic needle, 368. 

Magnetic repulsion, 361, 

Magnetic substances, 361. 

Magnetism, induced, 362; terres- 
trial, 368; theory of, 366. 

Magnifier, simple, 311. 

Major chord, 182. 

Major scale, 181. 

Mass, 3 ; center of, 65 ; measure- 
ment of, 9; unit of, 7. 

Matter, 2 ; indestructibility ot, 
3 ; states of, 11. 

Maxwell, sketch and portrait of, 
facing 432. 

Mechanical advantage, 85. 

Mechanical equivalent of heat, 
258. 

Melting point, 231; laws of, 232. 

Meter, standard, 4. 

Metric system, 4. 

Microphone, see Transmitter'. 

Microscope, 315. 

Mirrors, 278 ; concave, 284 ; con- 
vex, 285; images by, 281, 286, 
287; principal focus of, 284. 

Mixtures, method of, 227. 

Molecular energy, 211. 

Molecular forces, 130. 

Molecular theory of heat, 211. 

Molecules, 130. 

Moment of force, 89. 

Momentum, 28. 



478 



A HIGH SCHOOL COURSE IN PHYSICS 



Mont Blanc, 153. 

Morse, 453. 

Motion, 13; accelerated, 17; cir- 
cular, 44; compounded, 24, 29; 
first law of, 28; perpetual, 
259; rectilinear, 13; second law 
of, 30 ; third law of, 33 ; uni- 
form, 13. 

Motor, electric, 442; water, 115. 

Musical instruments, 197, 202. 

Musical scale, 180. 

Needle, dipping, 371; magnetic, 

368. 
Newton, sketch and portrait of, 

facing 30. 
NeM'ton's law of gravitation, 62. 
Newton's laws of motion, 28, 

30, 33. 
Nodes, in pipes, 201 ; in strings, 

194. 
Noise, 179. 
Non-conductors, 339. 

Octave, 182. 

Oersted, sketch and portrait of, 

facing 382. 
Ohm, sketch and portrait of, 

facing 406; unit, 406. 
Ohm's law, 408. 
Opaque bodies, 271. 
Opera glass, 316. 
Optical center, 308. 
Optical instruments, 311. 
Organ pipes, open, 199 ; overtones 

in, 201; stopped, 199. 
Oscillation, center of, 79. 
Oscillatory discharge, 462. 
Overtones, 194, 201. 

Parallel connection of cells, 411. 
Parallel forces, 40. 
Parallelogram of forces, 37. 
Partial tones, 194. 
Pascal, 112, 141. 
Pascal's law. 111. 
Pendulum, compound, 78; sim- 
ple, 74. 



Percussion, center of. 79. 

Period of vibration, 74. 

Permeability, 364. 

Phenomenon, 1. 

Phonograph, 204. 

Photometer, Bunsen's, 276. 

Physics, definition of, 1. 

Pigments, color of, 324; mixing, 
324. 

Pisa, tower of, 68, 69. 

Pitch, 179; of pipes, 199; of 
strings, 192; wave length and, 
173. 

Points, effect of, 344. 

Polarization of cells, 383. 

Poles, magnetic, 360. 

Potential, difference of, 347; fall 
of, 381; zero, 347. 

Potential energy, 55. 

Power, 53; candle, 276; electri- 
cal, 414. 

Pressure, in liquids, 103 ; of com- 
pressed air, 160; of gases, 148; 
of saturated vapor, 238; at- 
mospheric, 139. 

Primary coil, 429. 

Principle of Archimedes, 119. 

Prism, dispersion by, 321; re- 
fraction by, 298. 

Pulley, 85. 

Pump, air, 155; common lift, 
156; condensing, 156; force, 
158. 

Quality of sounds, 196. 

Radiant energy, 253; absorption 
of, 253. 

Radiation, 252 ; electrical, 463. 

Radio-activity, 468. 

Radiometer, 255. 

Radium, 468. 

Rainbow, 325. 

Ray, of light, 281; alpha, 469; 
beta, 469 ; cathode, 467 ; gam- 
ma, 469. 

Receiver, telephone, 457, 458. 



INDEX 



479 



Reeds, 204. 

Reflection, of light, 278; of 

sound, 176; angle of, 279; 

double, 282; law. of, 278; total, 

296. 
Refraction, 292; explained, 293; 

index of, 294; laws of, 293. 
Regelation, 233. 
Relay, 455. 
Resistance, of batteries, 409; of 

wires, 417; electrical, 409; 

laws of, 405; measurement of, 

418; unit of, 406. 
Resistance coils, 418. 
Resolution of forces, 42. 
Resonance, 185; explained, 187. 
Resultant, 21, 38. 
Retentivity, 362. 
Reversibility of pendulum, 79. 
Roemer, 269. 
Rontgen, sketch and portrait of, 

facing 466. 
Rontgen rays, 467. 
Rowland, 258. 
Rumford, sketch and portrait of, 

facing 256. 
Rutherford, 471. 

Sailboat, 43. 

Saturation, of vapors, 238. 

Saturation pressure, 238 ; mag- 
netic, 267. 

Scale, diatonic, 181 ; musical, 180. 

Screw, 97 ; mechanical advantage 
of, 98. 

Secondary coil, 429. 

Seconds pendulum, 78. 

Series connections, 410, 420. 

Shadows, 271. 

Sharps and flats, 183. 

Shunts, 421. 

Siphon, 159. » 

Siren, 180. 

Soap films, 130; color in, 332. 

Soddy, 471. 

Solar spectrum, 324 ; elements of, 
329. 



Solenoid, 390. 

Solids, density of, 124; sound in, 
168. 

Sonometer, 192. 

Sound, 165; air as a medium of, 
167; cause of, 165; intensity 
of, 173; interference of, 188; 
musical, 179; quality of, 196; 
reflection of, 176; speed of, 
168; wave motion of, 169. 

Sounder, telegraph, 453. 

Sounding' boards, 176. 

Spark, electric, 352, 356, 430, 
463. 

Specific gravity, 126. 

Specific heat, 226; measurement 
of, 227. 

Spectacles, 313. 

Spectra, 324; bright-line, 328; 
continuous, 327 ; elements iden- 
tified by, 329. 

Speed, of light, 269; of sound, 
168. 

Stability, 67. 

Stable equilibrium, 66. 

Steam engine, 260. 

Steam turbine, 263, 

Steelyard, 93. 

Stereoscope, 317. 

Storage cell, 399. 

Strength, current, 402. 

Strings, laws of, 192, 193. 

Substance, 3. 

Suction pump, 157. 

Sun, as source of energy, 255. 

Surface tension, 132. 

Suspension, point of, 79. 

Sympathetic vibrations, 185. 

Telegraph, 453; wireless, 464. 

Telephone, electric, 457 ; mechan- 
ical, 204. 

Telescope, 315; Galileo's, 316. 

Temperament, 184. 

Temperature, 208 ; absolute, 222 ; 
low, 223 ; measurement of, 212 ; 
scales of, 213. 



480 



A HIGH SCHOOL COURSE IN PHYSICS 



Tempered scales, 184. 

Tempering, 184. 

Thermal capacity, 226. 

Thermometer, centigrade, 213; 
comparison of, 214; dial, 218; 
Fahrenheit, 214; fixed points 
of, 212; Galileo's air, 215; 
graduation of, 213; range of, 
215. 

Thunder, 345. 

Time, unit of, 11. 

Toeppler-Holtz electrical ma- 
chine, 354. 

Torricelli's experiment, 141. 

Transformation of energy, 58. 

Transformer, 445. 

Transmission, of electricity, 415; 
of heat, 247; of light, 268; of 
sound, 167. 

Transmitter, 458. 

Transparent bodies, color of, 323. 

Transverse waves, 170. 

Tuning fork, 185. 

Turbine, water, 116; steam, 263. 

Units, of area, 6; of candle pow- 
er, 276; of capacity, 6; of cur- 
rent, 402; of electricity, 348; 
of force, 30; of heat, 225; of 
length, 4; of mass, 7; of po- 
tential, 407; of power, 53; of 
resistance, 406; of time, 11; 
of volume, 6; of work, 52. 

Vacuum, Torricellian, 142. 

Vapor, 237; saturated, 238. 

Vapor pressure, 238. 

Vaporization, 243; heat of, 249. 

Vector, 16. 

Velocity, 14; at any instant, 15; 
of falling bodies, 71; of light, 
269; of sound, 168; average, 
15; composition of, 22, 24; 
representation of, 15. 



Ventilation, 451. 

Vibrating body, 165. 

Vibration, of pipes, 200; of 

strings, 192; sympathetic, 185. 
Vibration numbers, 181, 184. 
Volt, 407. 
Volta, 375; sketch and portrait 

of, facing 352. 
Voltaic cell, 375. 
Voltmeter, 407. 

Water, abnormal expansion of, 
220; city supply of, 114; den- 
sity of, 107; greatest density 
of, 221. 

Water wheels, 115. 

Water turbine, 116. 

Watt, 54, 415. 

Watt-hour, 449. 

Wave length, 173; of light, 322; 
equation of, 173. 

Waves, electrical, 463 ; kinds of, 
170; longitudinal and trans- 
verse, 170; sound, 171; trans- 
mission of, 171. 

Wedge, 98. 

Weighing devices, 10. 

Weight, 263; and mass, 9; law 
of, 63. 

Wheatstone's bridge, 419. 

Wheel and axle, 93; mechanical 
advantage of, 94. 

White light, 321. 

Windlass, 94. 

Wireless telegraphy, 464. 

Work, 51; principle of, 83; rate 
of, 53; units of, 52. 

X-rays, 467; properties of, 468. 

Yard, metric equivalent of, 5. 

Zero potential, 347. 



(1) 



^y p,B 191 



One copy del. to Cat. Div. 



